{"id":1991,"date":"2018-01-11T21:12:16","date_gmt":"2018-01-11T21:12:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/limits-at-infinity-and-asymptotes\/"},"modified":"2018-02-05T16:59:33","modified_gmt":"2018-02-05T16:59:33","slug":"limits-at-infinity-and-asymptotes","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/limits-at-infinity-and-asymptotes\/","title":{"raw":"4.6 Limits at Infinity and Asymptotes","rendered":"4.6 Limits at Infinity and Asymptotes"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Calculate the limit of a function as [latex]x[\/latex] increases or decreases without bound.<\/li>\r\n \t<li>Recognize a horizontal asymptote on the graph of a function.<\/li>\r\n \t<li>Estimate the end behavior of a function as [latex]x[\/latex] increases or decreases without bound.<\/li>\r\n \t<li>Recognize an oblique asymptote on the graph of a function.<\/li>\r\n \t<li>Analyze a function and its derivatives to draw its graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function [latex]f[\/latex] defined on an unbounded domain, we also need to know the behavior of [latex]f[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function [latex]f.[\/latex]\r\n<div id=\"fs-id1165043145047\" class=\"bc-section section\">\r\n<h1>Limits at Infinity<\/h1>\r\n<p id=\"fs-id1165043308392\">We begin by examining what it means for a function to have a finite <strong>limit at infinity.<\/strong> Then we study the idea of a function with an <strong>infinite limit at infinity<\/strong>. Back in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction\/\">Introduction to Functions and Graphs<\/a>, we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.<\/p>\r\n\r\n<div id=\"fs-id1165042704801\" class=\"bc-section section\">\r\n<h2>Limits at Infinity and Horizontal Asymptotes<\/h2>\r\n<p id=\"fs-id1165043107285\">Recall that [latex]\\underset{x\\to a}{\\text{lim}}f(x)=L[\/latex] means [latex]f(x)[\/latex] becomes arbitrarily close to [latex]L[\/latex] as long as [latex]x[\/latex] is sufficiently close to [latex]a.[\/latex] We can extend this idea to limits at infinity. For example, consider the function [latex]f(x)=2+\\frac{1}{x}.[\/latex] As can be seen graphically in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_001\">(Figure)<\/a> and numerically in <a class=\"autogenerated-content\" href=\"#fs-id1165043428402\">(Figure)<\/a>, as the values of [latex]x[\/latex] get larger, the values of [latex]f(x)[\/latex] approach 2. We say the limit as [latex]x[\/latex] approaches [latex]\\infty [\/latex] of [latex]f(x)[\/latex] is 2 and write [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=2.[\/latex] Similarly, for [latex]x&lt;0,[\/latex] as the values [latex]|x|[\/latex] get larger, the values of [latex]f(x)[\/latex] approaches 2. We say the limit as [latex]x[\/latex] approaches [latex]\\text{\u2212}\\infty [\/latex] of [latex]f(x)[\/latex] is 2 and write [latex]\\underset{x\\to a}{\\text{lim}}f(x)=2.[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_06_001\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"717\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211025\/CNX_Calc_Figure_04_06_019.jpg\" alt=\"The function f(x) 2 + 1\/x is graphed. The function starts negative near y = 2 but then decreases to \u2212\u221e near x = 0. The function then decreases from \u221e near x = 0 and gets nearer to y = 2 as x increases. There is a horizontal line denoting the asymptote y = 2.\" width=\"717\" height=\"423\" \/> <strong>Figure 1.<\/strong> The function approaches the asymptote [latex]y=2[\/latex] as [latex]x[\/latex] approaches [latex]\\text{\u00b1}\\infty .[\/latex][\/caption]<\/div>\r\n<table id=\"fs-id1165043428402\" class=\"column-header\" summary=\"The table has four rows and five columns. The first column is a header column and it reads x, 2 + 1\/x, x, and 2 + 1\/x. After the header, the first row reads 10, 100, 1000, and 10000. The second row reads 2.1, 2.01, 2.001, and 2.0001. The third row reads \u221210, \u2212100, \u22121000, and \u221210000. The fourth row reads 1.9, 1.99, 1.999, and 1.9999.\"><caption>Values of a function [latex]f[\/latex] as [latex]x\\to \\text{\u00b1}\\infty [\/latex]<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<td>1,000<\/td>\r\n<td>10,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]2+\\frac{1}{x}[\/latex]<\/strong><\/td>\r\n<td>2.1<\/td>\r\n<td>2.01<\/td>\r\n<td>2.001<\/td>\r\n<td>2.0001<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>-10<\/td>\r\n<td>-100<\/td>\r\n<td>-1000<\/td>\r\n<td>-10,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]2+\\frac{1}{x}[\/latex]<\/strong><\/td>\r\n<td>1.9<\/td>\r\n<td>1.99<\/td>\r\n<td>1.999<\/td>\r\n<td>1.9999<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042936244\">More generally, for any function [latex]f,[\/latex] we say the limit as [latex]x\\to \\infty [\/latex] of [latex]f(x)[\/latex] is [latex]L[\/latex] if [latex]f(x)[\/latex] becomes arbitrarily close to [latex]L[\/latex] as long as [latex]x[\/latex] is sufficiently large. In that case, we write [latex]\\underset{x\\to a}{\\text{lim}}f(x)=L.[\/latex] Similarly, we say the limit as [latex]x\\to \\text{\u2212}\\infty [\/latex] of [latex]f(x)[\/latex] is [latex]L[\/latex] if [latex]f(x)[\/latex] becomes arbitrarily close to [latex]L[\/latex] as long as [latex]x&lt;0[\/latex] and [latex]|x|[\/latex] is sufficiently large. In that case, we write [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=L.[\/latex] We now look at the definition of a function having a limit at infinity.<\/p>\r\n\r\n<div id=\"fs-id1165042331960\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1165042970725\">(Informal) If the values of [latex]f(x)[\/latex] become arbitrarily close to [latex]L[\/latex] as [latex]x[\/latex] becomes sufficiently large, we say the function [latex]f[\/latex] has a limit at infinity and write<\/p>\r\n\r\n<div id=\"fs-id1165042986551\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L.[\/latex]<\/div>\r\n<p id=\"fs-id1165042374662\">If the values of [latex]f(x)[\/latex] becomes arbitrarily close to [latex]L[\/latex] for [latex]x&lt;0[\/latex] as [latex]|x|[\/latex] becomes sufficiently large, we say that the function [latex]f[\/latex] has a limit at negative infinity and write<\/p>\r\n\r\n<div id=\"fs-id1165043105208\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L.[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043157752\">If the values [latex]f(x)[\/latex] are getting arbitrarily close to some finite value [latex]L[\/latex] as [latex]x\\to \\infty [\/latex] or [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the graph of [latex]f[\/latex] approaches the line [latex]y=L.[\/latex] In that case, the line [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_002\">(Figure)<\/a>). For example, for the function [latex]f(x)=\\frac{1}{x},[\/latex] since [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=0,[\/latex] the line [latex]y=0[\/latex] is a horizontal asymptote of [latex]f(x)=\\frac{1}{x}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165043262534\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1165042973921\">If [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L[\/latex] or [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=L,[\/latex] we say the line [latex]y=L[\/latex] is a <strong>horizontal asymptote<\/strong> of [latex]f.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"CNX_Calc_Figure_04_06_002\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"766\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211028\/CNX_Calc_Figure_04_06_020.jpg\" alt=\"The figure is broken up into two figures labeled a and b. Figure a shows a function f(x) approaching but never touching a horizontal dashed line labeled L from above. Figure b shows a function f(x) approaching but never a horizontal dashed line labeled M from below.\" width=\"766\" height=\"273\" \/> <strong>Figure 2.<\/strong> (a) As [latex]x\\to \\infty ,[\/latex] the values of [latex]f[\/latex] are getting arbitrarily close to [latex]L.[\/latex] The line [latex]y=L[\/latex] is a horizontal asymptote of [latex]f.[\/latex] (b) As [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the values of [latex]f[\/latex] are getting arbitrarily close to [latex]M.[\/latex] The line [latex]y=M[\/latex] is a horizontal asymptote of [latex]f.[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165042647732\">A function cannot cross a vertical asymptote because the graph must approach infinity (or [latex]\\text{\u2212}\\infty )[\/latex] from at least one direction as [latex]x[\/latex] approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function [latex]f(x)=\\frac{( \\cos x)}{x}+1[\/latex] shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_003\">(Figure)<\/a> intersects the horizontal asymptote [latex]y=1[\/latex] an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_06_003\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"529\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211031\/CNX_Calc_Figure_04_06_002.jpg\" alt=\"The function f(x) = (cos x)\/x + 1 is shown. It decreases from (0, \u221e) and then proceeds to oscillate around y = 1 with decreasing amplitude.\" width=\"529\" height=\"230\" \/> <strong>Figure 3.<\/strong> The graph of [latex]f(x)=( \\cos x)\\text{\/}x+1[\/latex] crosses its horizontal asymptote [latex]y=1[\/latex] an infinite number of times.[\/caption]<\/div>\r\n<p id=\"fs-id1165042373486\">The algebraic limit laws and squeeze theorem we introduced in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-2\/\">Introduction to Limits<\/a> also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1165043429468\" class=\"textbox\">\r\n<h3>Computing Limits at Infinity<\/h3>\r\n<p id=\"fs-id1165043262623\">For each of the following functions [latex]f,[\/latex] evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x).[\/latex] Determine the horizontal asymptote(s) for [latex]f.[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1165042356111\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]f(x)=5-\\frac{2}{{x}^{2}}[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\frac{ \\sin x}{x}[\/latex]<\/li>\r\n \t<li>[latex]f(x)={ \\tan }^{-1}(x)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043183885\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043183885\"]\r\n<ol id=\"fs-id1165043183885\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Using the algebraic limit laws, we have [latex]\\underset{x\\to \\infty }{\\text{lim}}(5-\\frac{2}{{x}^{2}})=\\underset{x\\to \\infty }{\\text{lim}}5-2(\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{x}).(\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{x})=5-2\u00b70=5.[\/latex]Similarly, [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=5.[\/latex]Therefore, [latex]f(x)=5-\\frac{2}{{x}^{2}}[\/latex] has a horizontal asymptote of [latex]y=5[\/latex] and [latex]f[\/latex] approaches this horizontal asymptote as [latex]x\\to \\text{\u00b1}\\infty [\/latex] as shown in the following graph.\r\n<div id=\"CNX_Calc_Figure_04_06_004\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"492\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211033\/CNX_Calc_Figure_04_06_003.jpg\" alt=\"The function f(x) = 5 \u2013 2\/x2 is graphed. The function approaches the horizontal asymptote y = 5 as x approaches \u00b1\u221e.\" width=\"492\" height=\"309\" \/> <strong>Figure 4.<\/strong> This function approaches a horizontal asymptote as [latex]x\\to \\text{\u00b1}\\infty .[\/latex][\/caption]<\/div><\/li>\r\n \t<li>Since [latex]-1\\le \\sin x\\le 1[\/latex] for all [latex]x,[\/latex] we have\r\n<div id=\"fs-id1165043093355\" class=\"equation unnumbered\">[latex]\\frac{-1}{x}\\le \\frac{ \\sin x}{x}\\le \\frac{1}{x}[\/latex]<\/div>\r\nfor all [latex]x\\ne 0.[\/latex] Also, since\r\n<div id=\"fs-id1165043197153\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{-1}{x}=0=\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{x},[\/latex]<\/div>\r\nwe can apply the squeeze theorem to conclude that\r\n<div id=\"fs-id1165043036581\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{ \\sin x}{x}=0.[\/latex]<\/div>\r\nSimilarly,\r\n<div id=\"fs-id1165043122536\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{ \\sin x}{x}=0.[\/latex]<\/div>\r\n&nbsp;\r\n\r\nThus, [latex]f(x)=\\frac{ \\sin x}{x}[\/latex] has a horizontal asymptote of [latex]y=0[\/latex] and [latex]f(x)[\/latex] approaches this horizontal asymptote as [latex]x\\to \\text{\u00b1}\\infty [\/latex] as shown in the following graph.\r\n<div id=\"CNX_Calc_Figure_04_06_005\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"717\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211036\/CNX_Calc_Figure_04_06_004.jpg\" alt=\"The function f(x) = (sin x)\/x is shown. It has a global maximum at (0, 1) and then proceeds to oscillate around y = 0 with decreasing amplitude.\" width=\"717\" height=\"193\" \/> <strong>Figure 5.<\/strong> This function crosses its horizontal asymptote multiple times.[\/caption]\r\n\r\n<\/div><\/li>\r\n \t<li>To evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}{ \\tan }^{-1}(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{ \\tan }^{-1}(x),[\/latex] we first consider the graph of [latex]y= \\tan (x)[\/latex] over the interval [latex](\\text{\u2212}\\pi \\text{\/}2,\\pi \\text{\/}2)[\/latex] as shown in the following graph.\r\n<div id=\"CNX_Calc_Figure_04_06_006\" class=\"wp-caption aligncenter\"><span id=\"fs-id1165042710828\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211039\/CNX_Calc_Figure_04_06_021.jpg\" alt=\"The function f(x) = tan x is shown. It increases from (\u2212\u03c0\/2, \u2212\u221e), passes through the origin, and then increases toward (\u03c0\/2, \u221e). There are vertical dashed lines marking x = \u00b1\u03c0\/2.\" \/><\/span><\/div>\r\n<div class=\"wp-caption-text\">The graph of [latex] \\tan x[\/latex] has vertical asymptotes at [latex]x=\\text{\u00b1}\\frac{\\pi }{2}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165043092430\">Since<\/p>\r\n\r\n<div id=\"fs-id1165043119614\" class=\"equation unnumbered\">[latex]\\underset{x\\to {(\\pi \\text{\/}2)}^{-}}{\\text{lim}} \\tan x=\\infty ,[\/latex]<\/div>\r\n<p id=\"fs-id1165042514177\">it follows that<\/p>\r\n\r\n<div id=\"fs-id1165042563973\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}{ \\tan }^{-1}(x)=\\frac{\\pi }{2}.[\/latex]<\/div>\r\n<p id=\"fs-id1165043097156\">Similarly, since<\/p>\r\n\r\n<div id=\"fs-id1165042923290\" class=\"equation unnumbered\">[latex]\\underset{x\\to {(\\pi \\text{\/}2)}^{+}}{\\text{lim}} \\tan x=\\text{\u2212}\\infty ,[\/latex]<\/div>\r\n<p id=\"fs-id1165043131939\">it follows that<\/p>\r\n\r\n<div id=\"fs-id1165043056813\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{ \\tan }^{-1}(x)=-\\frac{\\pi }{2}.[\/latex]<\/div>\r\n<p id=\"fs-id1165042707528\">As a result, [latex]y=\\frac{\\pi }{2}[\/latex] and [latex]y=-\\frac{\\pi }{2}[\/latex] are horizontal asymptotes of [latex]f(x)={ \\tan }^{-1}(x)[\/latex] as shown in the following graph.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_06_007\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"491\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211043\/CNX_Calc_Figure_04_06_005.jpg\" alt=\"The function f(x) = tan\u22121 x is shown. It increases from (\u2212\u221e, \u2212\u03c0\/2), passes through the origin, and then increases toward (\u221e, \u03c0\/2). There are horizontal dashed lines marking y = \u00b1\u03c0\/2.\" width=\"491\" height=\"199\" \/> <strong>Figure 7.<\/strong> This function has two horizontal asymptotes.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042320881\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042320884\" class=\"exercise\">\r\n<div id=\"fs-id1165043315933\" class=\"textbox\">\r\n<p id=\"fs-id1165043315935\">Evaluate [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}(3+\\frac{4}{x})[\/latex] and [latex]\\underset{x\\to \\infty }{\\text{lim}}(3+\\frac{4}{x}).[\/latex] Determine the horizontal asymptotes of [latex]f(x)=3+\\frac{4}{x},[\/latex] if any.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043390798\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043390798\"]\r\n<p id=\"fs-id1165043390798\">Both limits are 3. The line [latex]y=3[\/latex] is a horizontal asymptote.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042318505\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042318511\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}1\\text{\/}x=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042333169\" class=\"bc-section section\">\r\n<h2>Infinite Limits at Infinity<\/h2>\r\n<p id=\"fs-id1165042333174\">Sometimes the values of a function [latex]f[\/latex] become arbitrarily large as [latex]x\\to \\infty [\/latex] (or as [latex]x\\to \\text{\u2212}\\infty ).[\/latex] In this case, we write [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty [\/latex] (or [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=\\infty ).[\/latex] On the other hand, if the values of [latex]f[\/latex] are negative but become arbitrarily large in magnitude as [latex]x\\to \\infty [\/latex] (or as [latex]x\\to \\text{\u2212}\\infty ),[\/latex] we write [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty [\/latex] (or [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty ).[\/latex]<\/p>\r\n<p id=\"fs-id1165042606820\">For example, consider the function [latex]f(x)={x}^{3}.[\/latex] As seen in <a class=\"autogenerated-content\" href=\"#fs-id1165042406634\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_008\">(Figure)<\/a>, as [latex]x\\to \\infty [\/latex] the values [latex]f(x)[\/latex] become arbitrarily large. Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{3}=\\infty .[\/latex] On the other hand, as [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the values of [latex]f(x)={x}^{3}[\/latex] are negative but become arbitrarily large in magnitude. Consequently, [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{3}=\\text{\u2212}\\infty .[\/latex]<\/p>\r\n\r\n<table id=\"fs-id1165042406634\" class=\"column-header\" summary=\"The table has four rows and six columns. The first column is a header column and it reads x, x3, x, and x3. After the header, the first row reads 10, 20, 50, 100, and 1000. The second row reads 1000, 8000, 125000, 1,000,000, and 1,000,000,000. The third row reads \u221210, \u221220, \u221250, \u2212100, and \u22121000. The forth row reads \u22121000, \u22128000, \u2212125,000, \u22121,000,000, and \u22121,000,000,000.\"><caption>Values of a power function as [latex]x\\to \\text{\u00b1}\\infty [\/latex]<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>20<\/td>\r\n<td>50<\/td>\r\n<td>100<\/td>\r\n<td>1000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]{x}^{3}[\/latex]<\/strong><\/td>\r\n<td>1000<\/td>\r\n<td>8000<\/td>\r\n<td>125,000<\/td>\r\n<td>1,000,000<\/td>\r\n<td>1,000,000,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>-10<\/td>\r\n<td>-20<\/td>\r\n<td>-50<\/td>\r\n<td>-100<\/td>\r\n<td>-1000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]{x}^{3}[\/latex]<\/strong><\/td>\r\n<td>-1000<\/td>\r\n<td>-8000<\/td>\r\n<td>-125,000<\/td>\r\n<td>-1,000,000<\/td>\r\n<td>-1,000,000,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"CNX_Calc_Figure_04_06_008\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"642\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211045\/CNX_Calc_Figure_04_06_022.jpg\" alt=\"The function f(x) = x3 is graphed. It is apparent that this function rapidly approaches infinity as x approaches infinity.\" width=\"642\" height=\"272\" \/> <strong>Figure 8.<\/strong> For this function, the functional values approach infinity as [latex]x\\to \\text{\u00b1}\\infty .[\/latex][\/caption]<\/div>\r\n<div id=\"fs-id1165043276353\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1165043276356\">(Informal) We say a function [latex]f[\/latex] has an infinite limit at infinity and write<\/p>\r\n\r\n<div id=\"fs-id1165043276364\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty .[\/latex]<\/div>\r\n<p id=\"fs-id1165042709557\">if [latex]f(x)[\/latex] becomes arbitrarily large for [latex]x[\/latex] sufficiently large. We say a function has a negative infinite limit at infinity and write<\/p>\r\n\r\n<div id=\"fs-id1165042647077\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty .[\/latex]<\/div>\r\n<p id=\"fs-id1165042327355\">if [latex]f(x)&lt;0[\/latex] and [latex]|f(x)|[\/latex] becomes arbitrarily large for [latex]x[\/latex] sufficiently large. Similarly, we can define infinite limits as [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042328702\" class=\"bc-section section\">\r\n<h2>Formal Definitions<\/h2>\r\n<p id=\"fs-id1165042328707\">Earlier, we used the terms <em>arbitrarily close<\/em>, <em>arbitrarily large<\/em>, and <em>sufficiently large<\/em> to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.<\/p>\r\n\r\n<div id=\"fs-id1165043308442\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1165043308445\">(Formal) We say a function [latex]f[\/latex] has a limit at infinity, if there exists a real number [latex]L[\/latex] such that for all [latex]\\epsilon &gt;0,[\/latex] there exists [latex]N&gt;0[\/latex] such that<\/p>\r\n\r\n<div id=\"fs-id1165043395062\" class=\"equation unnumbered\">[latex]|f(x)-L|&lt;\\epsilon [\/latex]<\/div>\r\n<p id=\"fs-id1165043298558\">for all [latex]x&gt;N.[\/latex] In that case, we write<\/p>\r\n\r\n<div id=\"fs-id1165042364605\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L[\/latex]<\/div>\r\n<p id=\"fs-id1165042512686\">(see <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_009\">(Figure)<\/a>).<\/p>\r\n<p id=\"fs-id1165042327662\">We say a function [latex]f[\/latex] has a limit at negative infinity if there exists a real number [latex]L[\/latex] such that for all [latex]\\epsilon &gt;0,[\/latex] there exists [latex]N&lt;0[\/latex] such that<\/p>\r\n\r\n<div id=\"fs-id1165042331766\" class=\"equation unnumbered\">[latex]|f(x)-L|&lt;\\epsilon [\/latex]<\/div>\r\n<p id=\"fs-id1165042472034\">for all [latex]x&lt;N.[\/latex] In that case, we write<\/p>\r\n\r\n<div id=\"fs-id1165042472050\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=L.[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"CNX_Calc_Figure_04_06_009\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"369\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211049\/CNX_Calc_Figure_04_06_023.jpg\" alt=\"The function f(x) is graphed, and it has a horizontal asymptote at L. L is marked on the y axis, as is L + \u0949 and L \u2013 \u0949. On the x axis, N is marked as the value of x such that f(x) = L + \u0949.\" width=\"369\" height=\"278\" \/> <strong>Figure 9.<\/strong> For a function with a limit at infinity, for all [latex]x&gt;N,[\/latex] [latex]|f(x)-L|&lt;\\epsilon .[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165043396243\">Earlier in this section, we used graphical evidence in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_001\">(Figure)<\/a> and numerical evidence in <a class=\"autogenerated-content\" href=\"#fs-id1165043428402\">(Figure)<\/a> to conclude that [latex]\\underset{x\\to \\infty }{\\text{lim}}(\\frac{2+1}{x})=2.[\/latex] Here we use the formal definition of limit at infinity to prove this result rigorously.<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>A Finite Limit at Infinity Example<\/h3>\r\n<div id=\"fs-id1165042587292\" class=\"exercise\">\r\n<div id=\"fs-id1165042587294\" class=\"textbox\">\r\n<p id=\"fs-id1165042587296\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\text{lim}}(\\frac{2+1}{x})=2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042369578\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042369578\"]\r\n<p id=\"fs-id1165042369578\">Let [latex]\\epsilon &gt;0.[\/latex] Let [latex]N=\\frac{1}{\\epsilon }.[\/latex] Therefore, for all [latex]x&gt;N,[\/latex] we have<\/p>\r\n\r\n<div id=\"fs-id1165043312498\" class=\"equation unnumbered\">[latex]|2+\\frac{1}{x}-2|=|\\frac{1}{x}|=\\frac{1}{x}&lt;\\frac{1}{N}=\\epsilon \\text{.}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042480092\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042480095\" class=\"exercise\">\r\n<div id=\"fs-id1165042480097\" class=\"textbox\">\r\n<p id=\"fs-id1165042480099\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\text{lim}}(\\frac{3-1}{{x}^{2}})=3.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042367887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042367887\"]\r\n<p id=\"fs-id1165042367887\">Let [latex]\\epsilon &gt;0.[\/latex] Let [latex]N=\\frac{1}{\\sqrt{\\epsilon }}.[\/latex] Therefore, for all [latex]x&gt;N,[\/latex] we have<\/p>\r\n<p id=\"fs-id1165042376362\">[latex]|3-\\frac{1}{{x}^{2}}-3|=\\frac{1}{{x}^{2}}&lt;\\frac{1}{{N}^{2}}=\\epsilon [\/latex]<\/p>\r\n<p id=\"fs-id1165042320298\">Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}(3-1\\text{\/}{x}^{2})=3.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042332059\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042332065\">Let [latex]N=\\frac{1}{\\sqrt{\\epsilon }}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042374773\">We now turn our attention to a more precise definition for an infinite limit at infinity.<\/p>\r\n\r\n<div id=\"fs-id1165042374776\" class=\"textbox key-takeaways\">\r\n<div class=\"title\">\r\n<h3>Definition<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1165042374780\">(Formal) We say a function [latex]f[\/latex] has an infinite limit at infinity and write<\/p>\r\n\r\n<div id=\"fs-id1165042364247\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty [\/latex]<\/div>\r\n<p id=\"fs-id1165043423999\">if for all [latex]M&gt;0,[\/latex] there exists an [latex]N&gt;0[\/latex] such that<\/p>\r\n\r\n<div id=\"fs-id1165043248795\" class=\"equation unnumbered\">[latex]f(x)&gt;M[\/latex]<\/div>\r\n<p id=\"fs-id1165042374733\">for all [latex]x&gt;N[\/latex] (see <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_010\">(Figure)<\/a>).<\/p>\r\n<p id=\"fs-id1165042374750\">We say a function has a negative infinite limit at infinity and write<\/p>\r\n\r\n<div id=\"fs-id1165042374753\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty [\/latex]<\/div>\r\n<p id=\"fs-id1165043426267\">if for all [latex]M&lt;0,[\/latex] there exists an [latex]N&gt;0[\/latex] such that<\/p>\r\n\r\n<div id=\"fs-id1165043259687\" class=\"equation unnumbered\">[latex]f(x)&lt;M[\/latex]<\/div>\r\n<p id=\"fs-id1165043259707\">for all [latex]x&gt;N.[\/latex]<\/p>\r\n<p id=\"fs-id1165043259751\">Similarly we can define limits as [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"CNX_Calc_Figure_04_06_010\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"456\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211052\/CNX_Calc_Figure_04_06_024.jpg\" alt=\"The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M.\" width=\"456\" height=\"315\" \/> <strong>Figure 10.<\/strong> For a function with an infinite limit at infinity, for all [latex]x&gt;N,[\/latex] [latex]f(x)&gt;M.[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165042705963\">Earlier, we used graphical evidence (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_008\">(Figure)<\/a>) and numerical evidence (<a class=\"autogenerated-content\" href=\"#fs-id1165042406634\">(Figure)<\/a>) to conclude that [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{3}=\\infty .[\/latex] Here we use the formal definition of infinite limit at infinity to prove that result.<\/p>\r\n\r\n<div id=\"fs-id1165042323534\" class=\"textbox examples\">\r\n<h3>An Infinite Limit at Infinity<\/h3>\r\n<div id=\"fs-id1165042323536\" class=\"exercise\">\r\n<div id=\"fs-id1165042323538\" class=\"textbox\">\r\n<p id=\"fs-id1165043395589\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{3}=\\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043430975\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043430975\"]\r\n<p id=\"fs-id1165043430975\">Let [latex]M&gt;0.[\/latex] Let [latex]N=\\sqrt[3]{M}.[\/latex] Then, for all [latex]x&gt;N,[\/latex] we have<\/p>\r\n\r\n<div id=\"fs-id1165043174087\" class=\"equation unnumbered\">[latex]{x}^{3}&gt;{N}^{3}={(\\sqrt[3]{M})}^{3}=M.[\/latex]<\/div>\r\n<p id=\"fs-id1165042604681\">Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{3}=\\infty .[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042323710\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042320222\" class=\"exercise\">\r\n<div id=\"fs-id1165042320224\" class=\"textbox\">\r\n<p id=\"fs-id1165042320226\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\text{lim}}3{x}^{2}=\\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042708272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042708272\"]\r\n<p id=\"fs-id1165042708272\">Let [latex]M&gt;0.[\/latex] Let [latex]N=\\sqrt{\\frac{M}{3}}.[\/latex] Then, for all [latex]x&gt;N,[\/latex] we have<\/p>\r\n<p id=\"fs-id1165042383154\">[latex]3{x}^{2}&gt;3{N}^{2}=3{(\\sqrt{\\frac{M}{3}})}^{2}{2}^{}=\\frac{3M}{3}=M[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043219098\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043219104\">Let [latex]N=\\sqrt{\\frac{M}{3}}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042368487\" class=\"bc-section section\">\r\n<h1>End Behavior<\/h1>\r\n<p id=\"fs-id1165042368492\">The behavior of a function as [latex]x\\to \\text{\u00b1}\\infty [\/latex] is called the function\u2019s <strong>end behavior<\/strong>. At each of the function\u2019s ends, the function could exhibit one of the following types of behavior:<\/p>\r\n\r\n<ol id=\"fs-id1165042349939\">\r\n \t<li>The function [latex]f(x)[\/latex] approaches a horizontal asymptote [latex]y=L.[\/latex]<\/li>\r\n \t<li>The function [latex]f(x)\\to \\infty [\/latex] or [latex]f(x)\\to \\text{\u2212}\\infty .[\/latex]<\/li>\r\n \t<li>The function does not approach a finite limit, nor does it approach [latex]\\infty [\/latex] or [latex]\\text{\u2212}\\infty .[\/latex] In this case, the function may have some oscillatory behavior.<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165042323661\">Let\u2019s consider several classes of functions here and look at the different types of end behaviors for these functions.<\/p>\r\n\r\n<div id=\"fs-id1165042323666\" class=\"bc-section section\">\r\n<h2>End Behavior for Polynomial Functions<\/h2>\r\n<p id=\"fs-id1165042323672\">Consider the power function [latex]f(x)={x}^{n}[\/latex] where [latex]n[\/latex] is a positive integer. From <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_011\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_012\">(Figure)<\/a>, we see that<\/p>\r\n\r\n<div id=\"fs-id1165042545843\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}=\\infty ;n=1,2,3\\text{,\u2026}[\/latex]<\/div>\r\nand\r\n<div id=\"fs-id1165042705928\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}=\\bigg\\{\\begin{array}{c}\\infty ;n=2,4,6\\text{,\u2026}\\hfill \\\\ \\text{\u2212}\\infty ;n=1,3,5\\text{,\u2026}\\hfill \\end{array}.[\/latex]<\/div>\r\n<div id=\"CNX_Calc_Figure_04_06_011\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"425\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211055\/CNX_Calc_Figure_04_06_025.jpg\" alt=\"The functions x2, x4, and x6 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.\" width=\"425\" height=\"358\" \/> <strong>Figure 11.<\/strong> For power functions with an even power of [latex]n,[\/latex] [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}=\\infty =\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}.[\/latex][\/caption]<\/div>\r\n<div id=\"CNX_Calc_Figure_04_06_012\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211058\/CNX_Calc_Figure_04_06_026.jpg\" alt=\"The functions x, x3, and x5 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.\" width=\"417\" height=\"352\" \/> <strong>Figure 12.<\/strong> For power functions with an odd power of [latex]n,[\/latex] [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}=\\infty [\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}=\\text{\u2212}\\infty .[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165042318644\">Using these facts, it is not difficult to evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}c{x}^{n}[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}c{x}^{n},[\/latex] where [latex]c[\/latex] is any constant and [latex]n[\/latex] is a positive integer. If [latex]c&gt;0,[\/latex] the graph of [latex]y=c{x}^{n}[\/latex] is a vertical stretch or compression of [latex]y={x}^{n},[\/latex] and therefore<\/p>\r\n\r\n<div id=\"fs-id1165042327426\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}c{x}^{n}=\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}\\text{ and }\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}c{x}^{n}=\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}\\text{ if }c&gt;0.[\/latex]<\/div>\r\n<p id=\"fs-id1165043424818\">If [latex]c&lt;0,[\/latex] the graph of [latex]y=c{x}^{n}[\/latex] is a vertical stretch or compression combined with a reflection about the [latex]x[\/latex]-axis, and therefore<\/p>\r\n\r\n<div id=\"fs-id1165042327325\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}c{x}^{n}=\\text{\u2212}\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}\\text{ and }\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}c{x}^{n}=\\text{\u2212}\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}\\text{ if }c&lt;0.[\/latex]<\/div>\r\n<p id=\"fs-id1165042640745\">If [latex]c=0,y=c{x}^{n}=0,[\/latex] in which case [latex]\\underset{x\\to \\infty }{\\text{lim}}c{x}^{n}=0=\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}c{x}^{n}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165043219126\" class=\"textbox examples\">\r\n<h3>Limits at Infinity for Power Functions<\/h3>\r\n<div id=\"fs-id1165043219128\" class=\"exercise\">\r\n<div id=\"fs-id1165043219130\" class=\"textbox\">\r\n\r\nFor each function [latex]f,[\/latex] evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x).[\/latex]\r\n<ol id=\"fs-id1165042333246\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]f(x)=-5{x}^{3}[\/latex]<\/li>\r\n \t<li>[latex]f(x)=2{x}^{4}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043254252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043254252\"]\r\n<ol id=\"fs-id1165043254252\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Since the coefficient of [latex]{x}^{3}[\/latex] is -5, the graph of [latex]f(x)=-5{x}^{3}[\/latex] involves a vertical stretch and reflection of the graph of [latex]y={x}^{3}[\/latex] about the [latex]x[\/latex]-axis. Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}(-5{x}^{3})=\\text{\u2212}\\infty [\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}(-5{x}^{3})=\\infty .[\/latex]<\/li>\r\n \t<li>Since the coefficient of [latex]{x}^{4}[\/latex] is 2, the graph of [latex]f(x)=2{x}^{4}[\/latex] is a vertical stretch of the graph of [latex]y={x}^{4}.[\/latex] Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}2{x}^{4}=\\infty [\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}2{x}^{4}=\\infty .[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042401057\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042401060\" class=\"exercise\">\r\n<div id=\"fs-id1165042401062\" class=\"textbox\">\r\n<p id=\"fs-id1165042401064\">Let [latex]f(x)=-3{x}^{4}.[\/latex] Find [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042708212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042708212\"]\r\n<p id=\"fs-id1165042708212\">[latex]\\text{\u2212}\\infty [\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042708221\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042708228\">The coefficient -3 is negative.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042708240\">We now look at how the limits at infinity for power functions can be used to determine [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}f(x)[\/latex] for any polynomial function [latex]f.[\/latex] Consider a polynomial function<\/p>\r\n\r\n<div id=\"fs-id1165042710943\" class=\"equation unnumbered\">[latex]f(x)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\\text{\u2026}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\r\n<p id=\"fs-id1165043327638\">of degree [latex]n\\ge 1[\/latex] so that [latex]{a}_{n}\\ne 0.[\/latex] Factoring, we see that<\/p>\r\n\r\n<div id=\"fs-id1165042319257\" class=\"equation unnumbered\">[latex]f(x)={a}_{n}{x}^{n}(1+\\frac{{a}_{n-1}}{{a}_{n}}\\frac{1}{x}+\\text{\u2026}+\\frac{{a}_{1}}{{a}_{n}}\\frac{1}{{x}^{n-1}}+\\frac{{a}_{0}}{{a}_{n}}).[\/latex]<\/div>\r\n<p id=\"fs-id1165043348532\">As [latex]x\\to \\text{\u00b1}\\infty ,[\/latex] all the terms inside the parentheses approach zero except the first term. We conclude that<\/p>\r\n\r\n<div id=\"fs-id1165043348550\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}f(x)=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}{a}_{n}{x}^{n}.[\/latex]<\/div>\r\n<p id=\"fs-id1165043317360\">For example, the function [latex]f(x)=5{x}^{3}-3{x}^{2}+4[\/latex] behaves like [latex]g(x)=5{x}^{3}[\/latex] as [latex]x\\to \\text{\u00b1}\\infty [\/latex] as shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_013\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165043250976\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_06_013\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"382\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211101\/CNX_Calc_Figure_04_06_006.jpg\" alt=\"Both functions f(x) = 5x3 \u2013 3x2 + 4 and g(x) = 5x3 are plotted. Their behavior for large positive and large negative numbers converges.\" width=\"382\" height=\"272\" \/> <strong>Figure 13.<\/strong> The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.[\/caption]\r\n\r\n<\/div>\r\n<table id=\"fs-id1165043250976\" class=\"column-header\" summary=\"The table has six rows and four columns. The first column is a header column and it reads x, f(x) = 5x3 \u2013 3x2 + 4, g(x) = 5x3, x, f(x) = 5x3 \u2013 3x2 + 4, and g(x) = 5x3. After the header, the first row reads 10, 100, and 1000. The second row reads 4704, 4,970,004, and 4,997,000,004. The third row reads 5000, 5,000,000, 5,000,000,000. The fourth row reads \u221210, \u2212100, and \u22121000. The fifth row reads \u22125296, \u22125,029,996, and \u22125,002,999,996. The sixth row reads \u22125000, \u22125,000,000, and \u22125,000,000,000.\"><caption>A polynomial\u2019s end behavior is determined by the term with the largest exponent.<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<td>1000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]f(x)=5{x}^{3}-3{x}^{2}+4[\/latex]<\/strong><\/td>\r\n<td>4704<\/td>\r\n<td>4,970,004<\/td>\r\n<td>4,997,000,004<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]g(x)=5{x}^{3}[\/latex]<\/strong><\/td>\r\n<td>5000<\/td>\r\n<td>5,000,000<\/td>\r\n<td>5,000,000,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>-10<\/td>\r\n<td>-100<\/td>\r\n<td>-1000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]f(x)=5{x}^{3}-3{x}^{2}+4[\/latex]<\/strong><\/td>\r\n<td>-5296<\/td>\r\n<td>-5,029,996<\/td>\r\n<td>-5,002,999,996<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]g(x)=5{x}^{3}[\/latex]<\/strong><\/td>\r\n<td>-5000<\/td>\r\n<td>-5,000,000<\/td>\r\n<td>-5,000,000,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"bc-section section\">\r\n<h2>End Behavior for Algebraic Functions<\/h2>\r\n<p id=\"fs-id1165042638493\">The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In <a class=\"autogenerated-content\" href=\"#fs-id1165042638553\">(Figure)<\/a>, we show that the limits at infinity of a rational function [latex]f(x)=\\frac{p(x)}{q(x)}[\/latex] depend on the relationship between the degree of the numerator and the degree of the denominator. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of [latex]x[\/latex] appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of [latex]x.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165042638553\" class=\"textbox examples\">\r\n<h3>Determining End Behavior for Rational Functions<\/h3>\r\n<div id=\"fs-id1165042638555\" class=\"exercise\">\r\n<div id=\"fs-id1165042638557\" class=\"textbox\">\r\n<p id=\"fs-id1165042638562\">For each of the following functions, determine the limits as [latex]x\\to \\infty [\/latex] and [latex]x\\to \\text{\u2212}\\infty .[\/latex] Then, use this information to describe the end behavior of the function.<\/p>\r\n\r\n<ol id=\"fs-id1165043390828\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]f(x)=\\frac{3x-1}{2x+5}[\/latex] (<em>Note:<\/em> The degree of the numerator and the denominator are the same.)<\/li>\r\n \t<li>[latex]f(x)=\\frac{3{x}^{2}+2x}{4{x}^{3}-5x+7}[\/latex] (<em>Note:<\/em> The degree of numerator is less than the degree of the denominator.)<\/li>\r\n \t<li>[latex]f(x)=\\frac{3{x}^{2}+4x}{x+2}[\/latex] (<em>Note:<\/em> The degree of numerator is greater than the degree of the denominator.)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042708379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042708379\"]\r\n<ol id=\"fs-id1165042708379\" style=\"list-style-type: lower-alpha\">\r\n \t<li>The highest power of [latex]x[\/latex] in the denominator is [latex]x.[\/latex] Therefore, dividing the numerator and denominator by [latex]x[\/latex] and applying the algebraic limit laws, we see that\r\n<div id=\"fs-id1165043281584\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3x-1}{2x+5}&amp; =\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3-1\\text{\/}x}{2+5\\text{\/}x}\\hfill \\\\ &amp; =\\frac{\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}(3-1\\text{\/}x)}{\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}(2+5\\text{\/}x)}\\hfill \\\\ &amp; =\\frac{\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}3-\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}1\\text{\/}x}{\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}2+\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}5\\text{\/}x}\\hfill \\\\ &amp; =\\frac{3-0}{2+0}=\\frac{3}{2}.\\hfill \\end{array}[\/latex]<\/div>\r\nSince [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}f(x)=\\frac{3}{2},[\/latex] we know that [latex]y=\\frac{3}{2}[\/latex] is a horizontal asymptote for this function as shown in the following graph.\r\n<div id=\"CNX_Calc_Figure_04_06_014\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"342\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211104\/CNX_Calc_Figure_04_06_007.jpg\" alt=\"The function f(x) = (3x + 1)\/(2x + 5) is plotted as is its horizontal asymptote at y = 3\/2.\" width=\"342\" height=\"347\" \/> <strong>Figure 14.<\/strong> The graph of this rational function approaches a horizontal asymptote as [latex]x\\to \\text{\u00b1}\\infty .[\/latex][\/caption]<\/div><\/li>\r\n \t<li>Since the largest power of [latex]x[\/latex] appearing in the denominator is [latex]{x}^{3},[\/latex] divide the numerator and denominator by [latex]{x}^{3}.[\/latex] After doing so and applying algebraic limit laws, we obtain\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3{x}^{2}+2x}{4{x}^{3}-5x+7}=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3\\text{\/}x+2\\text{\/}{x}^{2}}{4-5\\text{\/}{x}^{2}+7\\text{\/}{x}^{3}}=\\frac{3.0+2.0}{4-5.0+7.0}=0.[\/latex]<\/div>\r\nTherefore [latex]f[\/latex] has a horizontal asymptote of [latex]y=0[\/latex] as shown in the following graph.\r\n<div id=\"CNX_Calc_Figure_04_06_015\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211107\/CNX_Calc_Figure_04_06_008.jpg\" alt=\"The function f(x) = (3x2 + 2x)\/(4x2 \u2013 5x + 7) is plotted as is its horizontal asymptote at y = 0.\" width=\"417\" height=\"422\" \/> <strong>Figure 15.<\/strong> The graph of this rational function approaches the horizontal asymptote [latex]y=0[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex][\/caption]<\/div><\/li>\r\n \t<li>Dividing the numerator and denominator by [latex]x,[\/latex] we have\r\n<div id=\"fs-id1165042333346\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3{x}^{2}+4x}{x+2}=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3x+4}{1+2\\text{\/}x}.[\/latex]<\/div>\r\nAs [latex]x\\to \\text{\u00b1}\\infty ,[\/latex] the denominator approaches 1. As [latex]x\\to \\infty ,[\/latex] the numerator approaches [latex]+\\infty .[\/latex] As [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the numerator approaches [latex]\\text{\u2212}\\infty .[\/latex] Therefore [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty ,[\/latex] whereas [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty [\/latex] as shown in the following figure.\r\n<div id=\"CNX_Calc_Figure_04_06_016\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"569\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211110\/CNX_Calc_Figure_04_06_027.jpg\" alt=\"The function f(x) = (3x2 + 4x)\/(x + 2) is plotted. It appears to have a diagonal asymptote as well as a vertical asymptote at x = \u22122.\" width=\"569\" height=\"497\" \/> <strong>Figure 16.<\/strong> As [latex]x\\to \\infty ,[\/latex] the values [latex]f(x)\\to \\infty .[\/latex] As [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the values [latex]f(x)\\to \\text{\u2212}\\infty .[\/latex][\/caption]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042660288\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042660292\" class=\"exercise\">\r\n<div id=\"fs-id1165042660294\" class=\"textbox\">\r\n<p id=\"fs-id1165042660296\">Evaluate [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3{x}^{2}+2x-1}{5{x}^{2}-4x+7}[\/latex] and use these limits to determine the end behavior of [latex]f(x)=\\frac{3{x}^{2}+2x-2}{5{x}^{2}-4x+7}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042374871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042374871\"]\r\n<p id=\"fs-id1165042374871\">[latex]\\frac{3}{5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042374882\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042374889\">Divide the numerator and denominator by [latex]{x}^{2}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042374905\">Before proceeding, consider the graph of [latex]f(x)=\\frac{(3{x}^{2}+4x)}{(x+2)}[\/latex] shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_017\">(Figure)<\/a>. As [latex]x\\to \\infty [\/latex] and [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the graph of [latex]f[\/latex] appears almost linear. Although [latex]f[\/latex] is certainly not a linear function, we now investigate why the graph of [latex]f[\/latex] seems to be approaching a linear function. First, using long division of polynomials, we can write<\/p>\r\n\r\n<div id=\"fs-id1165043219202\" class=\"equation unnumbered\">[latex]f(x)=\\frac{3{x}^{2}+4x}{x+2}=3x-2+\\frac{4}{x+2}.[\/latex]<\/div>\r\n<p id=\"fs-id1165043219268\">Since [latex]\\frac{4}{(x+2)}\\to 0[\/latex] as [latex]x\\to \\text{\u00b1}\\infty ,[\/latex] we conclude that<\/p>\r\n\r\n<div id=\"fs-id1165042465555\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}(f(x)-(3x-2))=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{4}{x+2}=0.[\/latex]<\/div>\r\n<p id=\"fs-id1165042465646\">Therefore, the graph of [latex]f[\/latex] approaches the line [latex]y=3x-2[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] This line is known as an <strong>oblique asymptote<\/strong> for [latex]f[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_017\">(Figure)<\/a>).<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_06_017\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"267\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211113\/CNX_Calc_Figure_04_06_009.jpg\" alt=\"The function f(x) = (3x2 + 4x)\/(x + 2) is plotted as is its diagonal asymptote y = 3x \u2013 2.\" width=\"267\" height=\"347\" \/> Figure 17. The graph of the rational function [latex]f(x)=(3{x}^{2}+4x)\\text{\/}(x+2)[\/latex] approaches the oblique asymptote [latex]y=3x-2\\text{as}x\\to \\text{\u00b1}\\infty .[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165042461217\">We can summarize the results of <a class=\"autogenerated-content\" href=\"#fs-id1165042638553\">(Figure)<\/a> to make the following conclusion regarding end behavior for rational functions. Consider a rational function<\/p>\r\n\r\n<div id=\"fs-id1165042461226\" class=\"equation unnumbered\">[latex]f(x)=\\frac{p(x)}{q(x)}=\\frac{{a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\\text{\u2026}+{a}_{1}x+{a}_{0}}{{b}_{m}{x}^{m}+{b}_{m-1}{x}^{m-1}+\\text{\u2026}+{b}_{1}x+{b}_{0}},[\/latex]<\/div>\r\n<p id=\"fs-id1165042315666\">where [latex]{a}_{n}\\ne 0\\text{ and }{b}_{m}\\ne 0.[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1165043422338\">\r\n \t<li>If the degree of the numerator is the same as the degree of the denominator [latex](n=m),[\/latex] then [latex]f[\/latex] has a horizontal asymptote of [latex]y={a}_{n}\\text{\/}{b}_{m}[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/li>\r\n \t<li>If the degree of the numerator is less than the degree of the denominator [latex](n&lt;m),[\/latex] then [latex]f[\/latex] has a horizontal asymptote of [latex]y=0[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/li>\r\n \t<li>If the degree of the numerator is greater than the degree of the denominator [latex](n&gt;m),[\/latex] then [latex]f[\/latex] does not have a horizontal asymptote. The limits at infinity are either positive or negative infinity, depending on the signs of the leading terms. In addition, using long division, the function can be rewritten as\r\n<div id=\"fs-id1165043422484\" class=\"equation unnumbered\">[latex]f(x)=\\frac{p(x)}{q(x)}=g(x)+\\frac{r(x)}{q(x)},[\/latex]<\/div>\r\nwhere the degree of [latex]r(x)[\/latex] is less than the degree of [latex]q(x).[\/latex] As a result, [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}r(x)\\text{\/}q(x)=0.[\/latex] Therefore, the values of [latex]\\left[f(x)-g(x)\\right][\/latex] approach zero as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] If the degree of [latex]p(x)[\/latex] is exactly one more than the degree of [latex]q(x)[\/latex] [latex](n=m+1),[\/latex] the function [latex]g(x)[\/latex] is a linear function. In this case, we call [latex]g(x)[\/latex] an oblique asymptote.\r\nNow let\u2019s consider the end behavior for functions involving a radical.<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165042631816\" class=\"textbox examples\">\r\n<h3>Determining End Behavior for a Function Involving a Radical<\/h3>\r\n<div id=\"fs-id1165042631818\" class=\"exercise\">\r\n<div id=\"fs-id1165042631820\" class=\"textbox\">\r\n<p id=\"fs-id1165042631826\">Find the limits as [latex]x\\to \\infty [\/latex] and [latex]x\\to \\text{\u2212}\\infty [\/latex] for [latex]f(x)=\\frac{3x-2}{\\sqrt{4{x}^{2}+5}}[\/latex] and describe the end behavior of [latex]f.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042631905\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042631905\"]\r\n<p id=\"fs-id1165042631905\">Let\u2019s use the same strategy as we did for rational functions: divide the numerator and denominator by a power of [latex]x.[\/latex] To determine the appropriate power of [latex]x,[\/latex] consider the expression [latex]\\sqrt{4{x}^{2}+5}[\/latex] in the denominator. Since<\/p>\r\n\r\n<div id=\"fs-id1165042418061\" class=\"equation unnumbered\">[latex]\\sqrt{4{x}^{2}+5}\\approx \\sqrt{4{x}^{2}}=2|x|[\/latex]<\/div>\r\n<p id=\"fs-id1165042418106\">for large values of [latex]x[\/latex] in effect [latex]x[\/latex] appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by [latex]|x|.[\/latex] Then, using the fact that [latex]|x|=x[\/latex] for [latex]x&gt;0,[\/latex] [latex]|x|=\\text{\u2212}x[\/latex] for [latex]x&lt;0,[\/latex] and [latex]|x|=\\sqrt{{x}^{2}}[\/latex] for all [latex]x,[\/latex] we calculate the limits as follows:<\/p>\r\n\r\n<div id=\"fs-id1165042418216\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\underset{x\\to \\infty }{\\text{lim}}\\frac{3x-2}{\\sqrt{4{x}^{2}+5}}&amp; =\\hfill &amp; \\underset{x\\to \\infty }{\\text{lim}}\\frac{(1\\text{\/}|x|)(3x-2)}{(1\\text{\/}|x|)\\sqrt{4{x}^{2}+5}}\\hfill \\\\ &amp; =\\hfill &amp; \\underset{x\\to \\infty }{\\text{lim}}\\frac{(1\\text{\/}x)(3x-2)}{\\sqrt{(1\\text{\/}{x}^{2})(4{x}^{2}+5)}}\\hfill \\\\ &amp; =\\hfill &amp; \\underset{x\\to \\infty }{\\text{lim}}\\frac{3-2\\text{\/}x}{\\sqrt{4+5\\text{\/}{x}^{2}}}=\\frac{3}{\\sqrt{4}}=\\frac{3}{2}\\hfill \\\\ \\hfill \\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{3x-2}{\\sqrt{4{x}^{2}+5}}&amp; =\\hfill &amp; \\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{(1\\text{\/}|x|)(3x-2)}{(1\\text{\/}|x|)\\sqrt{4{x}^{2}+5}}\\hfill \\\\ &amp; =\\hfill &amp; \\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{(-1\\text{\/}x)(3x-2)}{\\sqrt{(1\\text{\/}{x}^{2})(4{x}^{2}+5)}}\\hfill \\\\ &amp; =\\hfill &amp; \\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{-3+2\\text{\/}x}{\\sqrt{4+5\\text{\/}{x}^{2}}}=\\frac{-3}{\\sqrt{4}}=\\frac{-3}{2}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165042463819\">Therefore, [latex]f(x)[\/latex] approaches the horizontal asymptote [latex]y=\\frac{3}{2}[\/latex] as [latex]x\\to \\infty [\/latex] and the horizontal asymptote [latex]y=-\\frac{3}{2}[\/latex] as [latex]x\\to \\text{\u2212}\\infty [\/latex] as shown in the following graph.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_06_018\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"592\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211116\/CNX_Calc_Figure_04_06_010.jpg\" alt=\"The function f(x) = (3x \u2212 2)\/(the square root of the quantity (4x2 + 5)) is plotted. It has two horizontal asymptotes at y = \u00b13\/2, and it crosses y = \u22123\/2 before converging toward it from below.\" width=\"592\" height=\"197\" \/> <strong>Figure 18.<\/strong> This function has two horizontal asymptotes and it crosses one of the asymptotes.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042463914\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042463919\" class=\"exercise\">\r\n<div id=\"fs-id1165042463921\" class=\"textbox\">\r\n<p id=\"fs-id1165042463923\">Evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{\\sqrt{3{x}^{2}+4}}{x+6}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043327293\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043327293\"]\r\n<p id=\"fs-id1165043327293\">[latex]\\text{\u00b1}\\sqrt{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043327303\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043327311\">Divide the numerator and denominator by [latex]|x|.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043327330\" class=\"bc-section section\">\r\n<h2>Determining End Behavior for Transcendental Functions<\/h2>\r\n<p id=\"fs-id1165043327335\">The six basic trigonometric functions are periodic and do not approach a finite limit as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] For example, [latex] \\sin x[\/latex] oscillates between [latex]1\\text{ and }-1[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_019\">(Figure)<\/a>). The tangent function [latex]x[\/latex] has an infinite number of vertical asymptotes as [latex]x\\to \\text{\u00b1}\\infty ;[\/latex] therefore, it does not approach a finite limit nor does it approach [latex]\\text{\u00b1}\\infty [\/latex] as [latex]x\\to \\text{\u00b1}\\infty [\/latex] as shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_020\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_06_019\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211121\/CNX_Calc_Figure_04_06_011.jpg\" alt=\"The function f(x) = sin x is graphed.\" width=\"417\" height=\"197\" \/> <strong>Figure 19.<\/strong> The function [latex]f(x)= \\sin x[\/latex] oscillates between [latex]1\\text{ and }-1[\/latex] as [latex]x\\to \\text{\u00b1}\\infty [\/latex][\/caption]<\/div>\r\n<div id=\"CNX_Calc_Figure_04_06_020\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"500\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211123\/CNX_Calc_Figure_04_06_012.jpg\" alt=\"The function f(x) = tan x is graphed.\" width=\"500\" height=\"272\" \/> <strong>Figure 20.<\/strong> The function [latex]f(x)= \\tan x[\/latex] does not approach a limit and does not approach [latex]\\text{\u00b1}\\infty [\/latex] as [latex]x\\to \\text{\u00b1}\\infty [\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165042459472\">Recall that for any base [latex]b&gt;0,b\\ne 1,[\/latex] the function [latex]y={b}^{x}[\/latex] is an exponential function with domain [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] and range [latex](0,\\infty ).[\/latex] If [latex]b&gt;1,y={b}^{x}[\/latex] is increasing over [latex]`(\\text{\u2212}\\infty ,\\infty ).[\/latex] If [latex]0&lt;b&lt;1,[\/latex] [latex]y={b}^{x}[\/latex] is decreasing over [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] For the natural exponential function [latex]f(x)={e}^{x},[\/latex] [latex]e\\approx 2.718&gt;1.[\/latex] Therefore, [latex]f(x)={e}^{x}[\/latex] is increasing on [latex]`(\\text{\u2212}\\infty ,\\infty )[\/latex] and the range is [latex]`(0,\\infty ).[\/latex] The exponential function [latex]f(x)={e}^{x}[\/latex] approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty [\/latex] and approaches 0 as [latex]x\\to \\text{\u2212}\\infty [\/latex] as shown in <a class=\"autogenerated-content\" href=\"#fs-id1165042542966\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_021\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"fs-id1165042542966\" class=\"column-header\" summary=\"The table has two rows and six columns. The first column is a header column and it reads x and ex. After the header, the first row reads \u22125, \u22122, 0, 2, and 5. The second row reads 0.00674, 0.135, 1, 7.389, and 148.413.\"><caption>End behavior of the natural exponential function<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>-5<\/td>\r\n<td>-2<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]{e}^{x}[\/latex]<\/strong><\/td>\r\n<td>0.00674<\/td>\r\n<td>0.135<\/td>\r\n<td>1<\/td>\r\n<td>7.389<\/td>\r\n<td>148.413<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"CNX_Calc_Figure_04_06_021\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"267\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211126\/CNX_Calc_Figure_04_06_013.jpg\" alt=\"The function f(x) = ex is graphed.\" width=\"267\" height=\"234\" \/> <strong>Figure 21.<\/strong> The exponential function approaches zero as [latex]x\\to \\text{\u2212}\\infty [\/latex] and approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty .[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165043218124\">Recall that the natural logarithm function [latex]f(x)=\\text{ln}(x)[\/latex] is the inverse of the natural exponential function [latex]y={e}^{x}.[\/latex] Therefore, the domain of [latex]f(x)=\\text{ln}(x)[\/latex] is [latex](0,\\infty )[\/latex] and the range is [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] The graph of [latex]f(x)=\\text{ln}(x)[\/latex] is the reflection of the graph of [latex]y={e}^{x}[\/latex] about the line [latex]y=x.[\/latex] Therefore, [latex]\\text{ln}(x)\\to \\text{\u2212}\\infty [\/latex] as [latex]x\\to {0}^{+}[\/latex] and [latex]\\text{ln}(x)\\to \\infty [\/latex] as [latex]x\\to \\infty [\/latex] as shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_022\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165042460463\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"fs-id1165042460463\" class=\"column-header\" summary=\"The table has two rows and six columns. The first column is a header column and it reads x and ln(x). After the header, the first row reads 0.01, 0.1, 1, 10, and 100. The second row reads \u22124.605, \u22122.303, 0, 2.303, and 4.605.\"><caption>End behavior of the natural logarithm function<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>0.01<\/td>\r\n<td>0.1<\/td>\r\n<td>1<\/td>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]\\text{ln}(x)[\/latex]<\/strong><\/td>\r\n<td>-4.605<\/td>\r\n<td>-2.303<\/td>\r\n<td>0<\/td>\r\n<td>2.303<\/td>\r\n<td>4.605<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"CNX_Calc_Figure_04_06_022\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211128\/CNX_Calc_Figure_04_06_014.jpg\" alt=\"The function f(x) = ln(x) is graphed.\" width=\"417\" height=\"272\" \/> <strong>Figure 22.<\/strong> The natural logarithm function approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty .[\/latex][\/caption]<\/div>\r\n<div id=\"fs-id1165042469818\" class=\"textbox examples\">\r\n<h3>Determining End Behavior for a Transcendental Function<\/h3>\r\n<div id=\"fs-id1165042469820\" class=\"exercise\">\r\n<div id=\"fs-id1165042469822\" class=\"textbox\">\r\n<p id=\"fs-id1165042469828\">Find the limits as [latex]x\\to \\infty [\/latex] and [latex]x\\to \\text{\u2212}\\infty [\/latex] for [latex]f(x)=\\frac{(2+3{e}^{x})}{(7-5{e}^{x})}[\/latex] and describe the end behavior of [latex]f.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042711624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711624\"]\r\n<p id=\"fs-id1165042711624\">To find the limit as [latex]x\\to \\infty ,[\/latex] divide the numerator and denominator by [latex]{e}^{x}\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165042711652\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\underset{x\\to \\infty }{\\text{lim}}f(x)&amp; =\\underset{x\\to \\infty }{\\text{lim}}\\frac{2+3{e}^{x}}{7-5{e}^{x}}\\hfill \\\\ &amp; =\\underset{x\\to \\infty }{\\text{lim}}\\frac{(2\\text{\/}{e}^{x})+3}{(7\\text{\/}{e}^{x})-5}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165042499478\">As shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_021\">(Figure)<\/a>, [latex]{e}^{x}\\to \\infty [\/latex] as [latex]x\\to \\infty .[\/latex] Therefore,<\/p>\r\n\r\n<div id=\"fs-id1165042499514\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{2}{{e}^{x}}=0=\\underset{x\\to \\infty }{\\text{lim}}\\frac{7}{{e}^{x}}.[\/latex]<\/div>\r\n<p id=\"fs-id1165042499577\">We conclude that [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=-\\frac{3}{5},[\/latex] and the graph of [latex]f[\/latex] approaches the horizontal asymptote [latex]y=-\\frac{3}{5}[\/latex] as [latex]x\\to \\infty .[\/latex] To find the limit as [latex]x\\to \\text{\u2212}\\infty ,[\/latex] use the fact that [latex]{e}^{x}\\to 0[\/latex] as [latex]x\\to \\text{\u2212}\\infty [\/latex] to conclude that [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\frac{2}{7},[\/latex] and therefore the graph of approaches the horizontal asymptote [latex]y=\\frac{2}{7}[\/latex] as [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042711306\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042711310\" class=\"exercise\">\r\n<div id=\"fs-id1165042711312\" class=\"textbox\">\r\n<p id=\"fs-id1165042711314\">Find the limits as [latex]x\\to \\infty [\/latex] and [latex]x\\to \\text{\u2212}\\infty [\/latex] for [latex]f(x)=\\frac{(3{e}^{x}-4)}{(5{e}^{x}+2)}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042711402\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711402\"]\r\n<p id=\"fs-id1165042711402\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\frac{3}{5},[\/latex][latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042711473\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042711480\">[latex]\\underset{x\\to \\infty }{\\text{lim}}{e}^{x}=\\infty [\/latex] and [latex]\\underset{x\\to \\infty }{\\text{lim}}{e}^{x}=0.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042602932\" class=\"bc-section section\">\r\n<h1>Guidelines for Drawing the Graph of a Function<\/h1>\r\n<p id=\"fs-id1165042602938\">We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let\u2019s look at a general strategy to use when graphing any function.<\/p>\r\n\r\n<div id=\"fs-id1165042602944\" class=\"textbox key-takeaways problem-solving\">\r\n<h3>Problem-Solving Strategy: Drawing the Graph of a Function<\/h3>\r\n<p id=\"fs-id1165042602951\">Given a function [latex]f,[\/latex] use the following steps to sketch a graph of [latex]f\\text{:}[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1165042602969\">\r\n \t<li>Determine the domain of the function.<\/li>\r\n \t<li>Locate the [latex]x[\/latex]- and [latex]y[\/latex]-intercepts.<\/li>\r\n \t<li>Evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)[\/latex] to determine the end behavior. If either of these limits is a finite number [latex]L,[\/latex] then [latex]y=L[\/latex] is a horizontal asymptote. If either of these limits is [latex]\\infty [\/latex] or [latex]\\text{\u2212}\\infty ,[\/latex] determine whether [latex]f[\/latex] has an oblique asymptote. If [latex]f[\/latex] is a rational function such that [latex]f(x)=\\frac{p(x)}{q(x)},[\/latex] where the degree of the numerator is greater than the degree of the denominator, then [latex]f[\/latex] can be written as\r\n<div id=\"fs-id1165042617521\" class=\"equation unnumbered\">[latex]f(x)=\\frac{p(x)}{q(x)}=g(x)+\\frac{r(x)}{q(x)},[\/latex]<\/div>\r\nwhere the degree of [latex]r(x)[\/latex] is less than the degree of [latex]q(x).[\/latex] The values of [latex]f(x)[\/latex] approach the values of [latex]g(x)[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] If [latex]g(x)[\/latex] is a linear function, it is known as an <em>oblique asymptote<\/em>.<\/li>\r\n \t<li>Determine whether [latex]f[\/latex] has any vertical asymptotes.<\/li>\r\n \t<li>Calculate [latex]{f}^{\\prime }.[\/latex] Find all critical points and determine the intervals where [latex]f[\/latex] is increasing and where [latex]f[\/latex] is decreasing. Determine whether [latex]f[\/latex] has any local extrema.<\/li>\r\n \t<li>Calculate [latex]f\\text{\u2033}.[\/latex] Determine the intervals where [latex]f[\/latex] is concave up and where [latex]f[\/latex] is concave down. Use this information to determine whether [latex]f[\/latex] has any inflection points. The second derivative can also be used as an alternate means to determine or verify that [latex]f[\/latex] has a local extremum at a critical point.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1165042617762\">Now let\u2019s use this strategy to graph several different functions. We start by graphing a polynomial function.<\/p>\r\n\r\n<div id=\"fs-id1165042617768\" class=\"textbox examples\">\r\n<h3>Sketching a Graph of a Polynomial<\/h3>\r\n<div id=\"fs-id1165042617770\" class=\"exercise\">\r\n<div id=\"fs-id1165042617772\" class=\"textbox\">\r\n<p id=\"fs-id1165042707708\">Sketch a graph of [latex]f(x)={(x-1)}^{2}(x+2).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042707761\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042707761\"]\r\n<p id=\"fs-id1165042707761\">Step 1. Since [latex]f[\/latex] is a polynomial, the domain is the set of all real numbers.<\/p>\r\n<p id=\"fs-id1165042707768\">Step 2. When [latex]x=0,f(x)=2.[\/latex] Therefore, the [latex]y[\/latex]-intercept is [latex](0,2).[\/latex] To find the [latex]x[\/latex]-intercepts, we need to solve the equation [latex]{(x-1)}^{2}(x+2)=0,[\/latex] gives us the [latex]x[\/latex]-intercepts [latex](1,0)[\/latex] and [latex](-2,0)[\/latex]<\/p>\r\n<p id=\"fs-id1165042707901\">Step 3. We need to evaluate the end behavior of [latex]f.[\/latex] As [latex]x\\to \\infty ,[\/latex] [latex]{(x-1)}^{2}\\to \\infty [\/latex] and [latex](x+2)\\to \\infty .[\/latex] Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty .[\/latex] As [latex]x\\to \\text{\u2212}\\infty ,[\/latex] [latex]{(x-1)}^{2}\\to \\infty [\/latex] and [latex](x+2)\\to \\text{\u2212}\\infty .[\/latex] Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty .[\/latex] To get even more information about the end behavior of [latex]f,[\/latex] we can multiply the factors of [latex]f.[\/latex] When doing so, we see that<\/p>\r\n\r\n<div id=\"fs-id1165042525463\" class=\"equation unnumbered\">[latex]f(x)={(x-1)}^{2}(x+2)={x}^{3}-3x+2.[\/latex]<\/div>\r\n<p id=\"fs-id1165042525532\">Since the leading term of [latex]f[\/latex] is [latex]{x}^{3},[\/latex] we conclude that [latex]f[\/latex] behaves like [latex]y={x}^{3}[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/p>\r\n<p id=\"fs-id1165043382921\">Step 4. Since [latex]f[\/latex] is a polynomial function, it does not have any vertical asymptotes.<\/p>\r\n<p id=\"fs-id1165043382928\">Step 5. The first derivative of [latex]f[\/latex] is<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=3{x}^{2}-3.[\/latex]<\/div>\r\n<p id=\"fs-id1165043382970\">Therefore, [latex]f[\/latex] has two critical points: [latex]x=1,-1.[\/latex] Divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the three smaller intervals: [latex](\\text{\u2212}\\infty ,-1),[\/latex] [latex](-1,1),[\/latex] and [latex](1,\\infty ).[\/latex] Then, choose test points [latex]x=-2,[\/latex] [latex]x=0,[\/latex] and [latex]x=2[\/latex] from these intervals and evaluate the sign of [latex]{f}^{\\prime }(x)[\/latex] at each of these test points, as shown in the following table.<\/p>\r\n\r\n<table id=\"fs-id1165043383127\" class=\"unnumbered\" summary=\"This table has four rows and four columns. The first row is a header row, and it reads Interval, Test Point, Sign of Derivative f\u2019(x) = 3x2 \u2013 3 = 3(x \u2013 1)(x + 1), and Conclusion. Under the header row, the first column reads (\u2212\u221e, \u22121), (\u22121, 1), and (1, \u221e). The second column reads x = \u22122, x = 0, and x = 2. The third column reads (+)(\u2212)(\u2212) = +, (+)(\u2212)(+) = \u2212, and (+)(+)(+) = +. The fourth column reads f is increasing, f is decreasing, and f is increasing.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of Derivative [latex]f\\prime (x)=3{x}^{2}-3=3(x-1)(x+1)[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\\text{\u2212}\\infty ,-1)[\/latex]<\/td>\r\n<td>[latex]x=-2[\/latex]<\/td>\r\n<td>[latex](\\text{+})(\\text{\u2212})(\\text{\u2212})=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](-1,1)[\/latex]<\/td>\r\n<td>[latex]x=0[\/latex]<\/td>\r\n<td>[latex](\\text{+})(\\text{\u2212})(\\text{+})=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](1,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=2[\/latex]<\/td>\r\n<td>[latex](\\text{+})(\\text{+})(\\text{+})=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042539204\">From the table, we see that [latex]f[\/latex] has a local maximum at [latex]x=-1[\/latex] and a local minimum at [latex]x=1.[\/latex] Evaluating [latex]f(x)[\/latex] at those two points, we find that the local maximum value is [latex]f(-1)=4[\/latex] and the local minimum value is [latex]f(1)=0.[\/latex]<\/p>\r\n<p id=\"fs-id1165042539284\">Step 6. The second derivative of [latex]f[\/latex] is<\/p>\r\n\r\n<div id=\"fs-id1165042539292\" class=\"equation unnumbered\">[latex]f\\text{\u2033}(x)=6x.[\/latex]<\/div>\r\n<p id=\"fs-id1165042539318\">The second derivative is zero at [latex]x=0.[\/latex] Therefore, to determine the concavity of [latex]f,[\/latex] divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the smaller intervals [latex](\\text{\u2212}\\infty ,0)[\/latex] and [latex](0,\\infty ),[\/latex] and choose test points [latex]x=-1[\/latex] and [latex]x=1[\/latex] to determine the concavity of [latex]f[\/latex] on each of these smaller intervals as shown in the following table.<\/p>\r\n\r\n<table id=\"fs-id1165042381927\" class=\"unnumbered\" summary=\"This table has three rows and four columns. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019\u2019(x) = 6x, and Conclusion. Under the header row, the first column reads (\u2212\u221e, 0) and (0, \u221e). The second column reads x = \u22121 and x = 1. The third column reads \u2212 and +. The fourth column reads f is concave down and f is concave up.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]f\\text{\u2033}(x)=6x[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\\text{\u2212}\\infty ,0)[\/latex]<\/td>\r\n<td>[latex]x=-1[\/latex]<\/td>\r\n<td>[latex]-[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave down.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](0,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=1[\/latex]<\/td>\r\n<td>[latex]+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave up.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042382153\">We note that the information in the preceding table confirms the fact, found in step 5, that [latex]f[\/latex] has a local maximum at [latex]x=-1[\/latex] and a local minimum at [latex]x=1.[\/latex] In addition, the information found in step 5\u2014namely, [latex]f[\/latex] has a local maximum at [latex]x=-1[\/latex] and a local minimum at [latex]x=1,[\/latex] and [latex]{f}^{\\prime }(x)=0[\/latex] at those points\u2014combined with the fact that [latex]f\\text{\u2033}[\/latex] changes sign only at [latex]x=0[\/latex] confirms the results found in step 6 on the concavity of [latex]f.[\/latex]<\/p>\r\n<p id=\"fs-id1165042709981\">Combining this information, we arrive at the graph of [latex]f(x)={(x-1)}^{2}(x+2)[\/latex] shown in the following graph.<\/p>\r\n<span id=\"fs-id1165042710031\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211132\/CNX_Calc_Figure_04_06_015.jpg\" alt=\"The function f(x) = (x \u22121)2 (x + 2) is graphed. It crosses the x axis at x = \u22122 and touches the x axis at x = 1.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042710046\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042710050\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1165042710055\">Sketch a graph of [latex]f(x)={(x-1)}^{3}(x+2).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042710106\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042710106\"]<span id=\"fs-id1165042710110\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211136\/CNX_Calc_Figure_04_06_028.jpg\" alt=\"The function f(x) = (x \u22121)3(x + 2) is graphed.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165042710122\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042710130\">[latex]f[\/latex] is a fourth-degree polynomial.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042710140\" class=\"textbox examples\">\r\n<h3>Sketching a Rational Function<\/h3>\r\n<div id=\"fs-id1165042710142\" class=\"exercise\">\r\n<div id=\"fs-id1165042710144\" class=\"textbox\">\r\n<p id=\"fs-id1165042710150\">Sketch the graph of [latex]f(x)=\\frac{{x}^{2}}{(1-{x}^{2})}\\text{.}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042407414\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042407414\"]\r\n<p id=\"fs-id1165042407414\">Step 1. The function [latex]f[\/latex] is defined as long as the denominator is not zero. Therefore, the domain is the set of all real numbers [latex]x[\/latex] except [latex]x=\\text{\u00b1}1.[\/latex]<\/p>\r\n<p id=\"fs-id1165042407440\">Step 2. Find the intercepts. If [latex]x=0,[\/latex] then [latex]f(x)=0,[\/latex] so 0 is an intercept. If [latex]y=0,[\/latex] then [latex]\\frac{{x}^{2}}{(1-{x}^{2})}=0,[\/latex] which implies [latex]x=0.[\/latex] Therefore, [latex](0,0)[\/latex] is the only intercept.<\/p>\r\n<p id=\"fs-id1165042407553\">Step 3. Evaluate the limits at infinity. Since [latex]f[\/latex] is a rational function, divide the numerator and denominator by the highest power in the denominator: [latex]{x}^{2}.[\/latex] We obtain<\/p>\r\n\r\n<div id=\"fs-id1165042407571\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{1}{\\frac{1}{{x}^{2}}-1}=-1.[\/latex]<\/div>\r\n<p id=\"fs-id1165042407655\">Therefore, [latex]f[\/latex] has a horizontal asymptote of [latex]y=-1[\/latex] as [latex]x\\to \\infty [\/latex] and [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\r\n<p id=\"fs-id1165042491004\">Step 4. To determine whether [latex]f[\/latex] has any vertical asymptotes, first check to see whether the denominator has any zeroes. We find the denominator is zero when [latex]x=\\text{\u00b1}1.[\/latex] To determine whether the lines [latex]x=1[\/latex] or [latex]x=-1[\/latex] are vertical asymptotes of [latex]f,[\/latex] evaluate [latex]\\underset{x\\to 1}{\\text{lim}}f(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}1}{\\text{lim}}f(x).[\/latex] By looking at each one-sided limit as [latex]x\\to 1,[\/latex] we see that<\/p>\r\n\r\n<div id=\"fs-id1165042491126\" class=\"equation unnumbered\">[latex]\\underset{x\\to {1}^{+}}{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\text{\u2212}\\infty \\text{ and }\\underset{x\\to {1}^{-}}{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\infty .[\/latex]<\/div>\r\n<p id=\"fs-id1165042491222\">In addition, by looking at each one-sided limit as [latex]x\\to \\text{\u2212}1,[\/latex] we find that<\/p>\r\n\r\n<div id=\"fs-id1165042491242\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}{1}^{+}}{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\infty \\text{ and }\\underset{x\\to \\text{\u2212}{1}^{-}}{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\text{\u2212}\\infty .[\/latex]<\/div>\r\n<p id=\"fs-id1165043262266\">Step 5. Calculate the first derivative:<\/p>\r\n\r\n<div id=\"fs-id1165043262270\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{(1-{x}^{2})(2x)-{x}^{2}(-2x)}{{(1-{x}^{2})}^{2}}=\\frac{2x}{{(1-{x}^{2})}^{2}}.[\/latex]<\/div>\r\n<p id=\"fs-id1165043262390\">Critical points occur at points [latex]x[\/latex] where [latex]{f}^{\\prime }(x)=0[\/latex] or [latex]{f}^{\\prime }(x)[\/latex] is undefined. We see that [latex]{f}^{\\prime }(x)=0[\/latex] when [latex]x=0.[\/latex] The derivative [latex]{f}^{\\prime }[\/latex] is not undefined at any point in the domain of [latex]f.[\/latex] However, [latex]x=\\text{\u00b1}1[\/latex] are not in the domain of [latex]f.[\/latex] Therefore, to determine where [latex]f[\/latex] is increasing and where [latex]f[\/latex] is decreasing, divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into four smaller intervals: [latex](\\text{\u2212}\\infty ,-1),[\/latex] [latex](-1,0),[\/latex] [latex](0,1),[\/latex] and [latex](1,\\infty ),[\/latex] and choose a test point in each interval to determine the sign of [latex]{f}^{\\prime }(x)[\/latex] in each of these intervals. The values [latex]x=-2,[\/latex] [latex]x=-\\frac{1}{2},[\/latex] [latex]x=\\frac{1}{2},[\/latex] and [latex]x=2[\/latex] are good choices for test points as shown in the following table.<\/p>\r\n\r\n<table id=\"fs-id1165043341618\" class=\"unnumbered\" summary=\"This table has four columns and five rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019(x) = 2x\/(1 \u2212 x2)2, and Conclusion. Under the header row, the first column reads (\u2212\u221e, \u22121), (\u22121, 0), (0, 1), and (1, \u221e). The second column reads x = \u22122, x = \u22121\/2, x = 1\/2, and x = 2. The third column reads \u2212\/+ = \u2212, \u2212\/+ = \u2212, +\/+ = +, and +\/+ = +. The fourth column reads f is decreasing, f is decreasing, f is increasing, and f is increasing.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]{f}^{\\prime }(x)=\\frac{2x}{{(1-{x}^{2})}^{2}}[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\\text{\u2212}\\infty ,-1)[\/latex]<\/td>\r\n<td>[latex]x=-2[\/latex]<\/td>\r\n<td>[latex]\\text{\u2212}\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](-1,0)[\/latex]<\/td>\r\n<td>[latex]x=-1\\text{\/}2[\/latex]<\/td>\r\n<td>[latex]\\text{\u2212}\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](0,1)[\/latex]<\/td>\r\n<td>[latex]x=1\\text{\/}2[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](1,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=2[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165043183348\">From this analysis, we conclude that [latex]f[\/latex] has a local minimum at [latex]x=0[\/latex] but no local maximum.<\/p>\r\n<p id=\"fs-id1165043183355\">Step 6. Calculate the second derivative:<\/p>\r\n\r\n<div id=\"fs-id1165043183358\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill f\\text{\u2033}(x)&amp; \\hfill =\\frac{{(1-{x}^{2})}^{2}(2)-2x(2(1-{x}^{2})(-2x))}{{(1-{x}^{2})}^{4}}\\\\ &amp; =\\frac{(1-{x}^{2})\\left[2(1-{x}^{2})+8{x}^{2}\\right]}{{(1-{x}^{2})}^{4}}\\hfill \\\\ &amp; =\\frac{2(1-{x}^{2})+8{x}^{2}}{{(1-{x}^{2})}^{3}}\\hfill \\\\ &amp; =\\frac{6{x}^{2}+2}{{(1-{x}^{2})}^{3}}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165043384128\">To determine the intervals where [latex]f[\/latex] is concave up and where [latex]f[\/latex] is concave down, we first need to find all points [latex]x[\/latex] where [latex]f\\text{\u2033}(x)=0[\/latex] or [latex]f\\text{\u2033}(x)[\/latex] is undefined. Since the numerator [latex]6{x}^{2}+2\\ne 0[\/latex] for any [latex]x,[\/latex] [latex]f\\text{\u2033}(x)[\/latex] is never zero. Furthermore, [latex]f\\text{\u2033}[\/latex] is not undefined for any [latex]x[\/latex] in the domain of [latex]f.[\/latex] However, as discussed earlier, [latex]x=\\text{\u00b1}1[\/latex] are not in the domain of [latex]f.[\/latex] Therefore, to determine the concavity of [latex]f,[\/latex] we divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the three smaller intervals [latex](\\text{\u2212}\\infty ,-1),[\/latex] [latex](-1,-1),[\/latex] and [latex](1,\\infty ),[\/latex] and choose a test point in each of these intervals to evaluate the sign of [latex]f\\text{\u2033}(x).[\/latex] in each of these intervals. The values [latex]x=-2,[\/latex] [latex]x=0,[\/latex] and [latex]x=2[\/latex] are possible test points as shown in the following table.<\/p>\r\n\r\n<table id=\"fs-id1165043384406\" class=\"unnumbered\" summary=\"This table has four columns and four rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019\u2019(x) = (6x2 + 2)\/(1 \u2212 x2)3, and Conclusion. Under the header row, the first column reads (\u2212\u221e, \u22121), (\u22121, 1), and (1, \u221e). The second column reads x = \u22122, x = 0, and x = 2. The third column reads +\/\u2212 = \u2212, +\/+ = +, and +\/\u2212 = \u2212. The fourth column reads f is concave down, f is concave up, and f is concave down.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]f\\text{\u2033}(x)=\\frac{6{x}^{2}+2}{{(1-{x}^{2})}^{3}}[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\\text{\u2212}\\infty ,-1)[\/latex]<\/td>\r\n<td>[latex]x=-2[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}-=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave down.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](-1,-1)[\/latex]<\/td>\r\n<td>[latex]x=0[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave up.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](1,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=2[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}-=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave down.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042592688\">Combining all this information, we arrive at the graph of [latex]f[\/latex] shown below. Note that, although [latex]f[\/latex] changes concavity at [latex]x=-1[\/latex] and [latex]x=1,[\/latex] there are no inflection points at either of these places because [latex]f[\/latex] is not continuous at [latex]x=-1[\/latex] or [latex]x=1.[\/latex]<\/p>\r\n<span id=\"fs-id1165042592749\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211139\/CNX_Calc_Figure_04_06_016.jpg\" alt=\"The function f(x) = x2\/(1 \u2212 x2) is graphed. It has asymptotes y = \u22121, x = \u22121, and x = 1.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042592764\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042592768\" class=\"exercise\">\r\n<div id=\"fs-id1165042592770\" class=\"textbox\">\r\n<p id=\"fs-id1165042592772\">Sketch a graph of [latex]f(x)=\\frac{(3x+5)}{(8+4x)}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042592825\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042592825\"]<span id=\"fs-id1165042592830\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211142\/CNX_Calc_Figure_04_06_029.jpg\" alt=\"The function f(x) = (3x + 5)\/(8 + 4x) is graphed. It appears to have asymptotes at x = \u22122 and y = 1.\" \/><\/span><\/div>\r\n<div id=\"fs-id1165042592842\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042592850\">A line [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex] if the limit as [latex]x\\to \\infty [\/latex] or the limit as [latex]x\\to \\text{\u2212}\\infty [\/latex] of [latex]f(x)[\/latex] is [latex]L.[\/latex] A line [latex]x=a[\/latex] is a vertical asymptote if at least one of the one-sided limits of [latex]f[\/latex] as [latex]x\\to a[\/latex] is [latex]\\infty [\/latex] or [latex]\\text{\u2212}\\infty .[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042592955\" class=\"textbox examples\">\r\n<h3>Sketching a Rational Function with an Oblique Asymptote<\/h3>\r\n<div id=\"fs-id1165042592957\" class=\"exercise\">\r\n<div id=\"fs-id1165042592959\" class=\"textbox\">\r\n<p id=\"fs-id1165042592964\">Sketch the graph of [latex]f(x)=\\frac{{x}^{2}}{(x-1)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042593006\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042593006\"]\r\n<p id=\"fs-id1165042593006\">Step 1. The domain of [latex]f[\/latex] is the set of all real numbers [latex]x[\/latex] except [latex]x=1.[\/latex]<\/p>\r\n<p id=\"fs-id1165042712534\">Step 2. Find the intercepts. We can see that when [latex]x=0,[\/latex] [latex]f(x)=0,[\/latex] so [latex](0,0)[\/latex] is the only intercept.<\/p>\r\n<p id=\"fs-id1165042712584\">Step 3. Evaluate the limits at infinity. Since the degree of the numerator is one more than the degree of the denominator, [latex]f[\/latex] must have an oblique asymptote. To find the oblique asymptote, use long division of polynomials to write<\/p>\r\n\r\n<div id=\"fs-id1165042712594\" class=\"equation unnumbered\">[latex]f(x)=\\frac{{x}^{2}}{x-1}=x+1+\\frac{1}{x-1}.[\/latex]<\/div>\r\n<p id=\"fs-id1165042712649\">Since [latex]1\\text{\/}(x-1)\\to 0[\/latex] as [latex]x\\to \\text{\u00b1}\\infty ,[\/latex] [latex]f(x)[\/latex] approaches the line [latex]y=x+1[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] The line [latex]y=x+1[\/latex] is an oblique asymptote for [latex]f.[\/latex]<\/p>\r\n<p id=\"fs-id1165042712758\">Step 4. To check for vertical asymptotes, look at where the denominator is zero. Here the denominator is zero at [latex]x=1.[\/latex] Looking at both one-sided limits as [latex]x\\to 1,[\/latex] we find<\/p>\r\n\r\n<div id=\"fs-id1165042712789\" class=\"equation unnumbered\">[latex]\\underset{x\\to {1}^{+}}{\\text{lim}}\\frac{{x}^{2}}{x-1}=\\infty \\text{ and }\\underset{x\\to {1}^{-}}{\\text{lim}}\\frac{{x}^{2}}{x-1}=\\text{\u2212}\\infty .[\/latex]<\/div>\r\n<p id=\"fs-id1165042403315\">Therefore, [latex]x=1[\/latex] is a vertical asymptote, and we have determined the behavior of [latex]f[\/latex] as [latex]x[\/latex] approaches 1 from the right and the left.<\/p>\r\n<p id=\"fs-id1165042403340\">Step 5. Calculate the first derivative:<\/p>\r\n\r\n<div id=\"fs-id1165042403344\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{(x-1)(2x)-{x}^{2}(1)}{{(x-1)}^{2}}=\\frac{{x}^{2}-2x}{{(x-1)}^{2}}.[\/latex]<\/div>\r\n<p id=\"fs-id1165042403459\">We have [latex]{f}^{\\prime }(x)=0[\/latex] when [latex]{x}^{2}-2x=x(x-2)=0.[\/latex] Therefore, [latex]x=0[\/latex] and [latex]x=2[\/latex] are critical points. Since [latex]f[\/latex] is undefined at [latex]x=1,[\/latex] we need to divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the smaller intervals [latex](\\text{\u2212}\\infty ,0),[\/latex] [latex](0,1),[\/latex] [latex](1,2),[\/latex] and [latex](2,\\infty ),[\/latex] and choose a test point from each interval to evaluate the sign of [latex]{f}^{\\prime }(x)[\/latex] in each of these smaller intervals. For example, let [latex]x=-1,[\/latex] [latex]x=\\frac{1}{2},[\/latex] [latex]x=\\frac{3}{2},[\/latex] and [latex]x=3[\/latex] be the test points as shown in the following table.<\/p>\r\n\r\n<table id=\"fs-id1165042710566\" class=\"unnumbered\" summary=\"This table has four columns and five rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019(x) = (x2 \u2212 2x)\/(x \u2212 1)2 = x(x \u2212 2)\/(x \u2212 1)2, and Conclusion. Under the header row, the first column reads (\u2212\u221e, 0), (0, 1), (1, 2), and (2, \u221e). The second column reads x = \u22121, x = 1\/2, x = 3\/2, and x = 3. The third column reads (\u2212)(\u2212)\/+ = +, (+)(\u2212)\/+ = \u2212, (+)(\u2212)\/+ = \u2212, and (+)(+)\/+ = +. The fourth column reads f is increasing, f is decreasing, f is decreasing, and f is increasing.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]f\\prime (x)=\\frac{{x}^{2}-2x}{{(x-1)}^{2}}=\\frac{x(x-2)}{{(x-1)}^{2}}[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\\text{\u2212}\\infty ,0)[\/latex]<\/td>\r\n<td>[latex]x=-1[\/latex]<\/td>\r\n<td>[latex](\\text{\u2212})(\\text{\u2212})\\text{\/}+=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](0,1)[\/latex]<\/td>\r\n<td>[latex]x=1\\text{\/}2[\/latex]<\/td>\r\n<td>[latex](\\text{+})(\\text{\u2212})\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](1,2)[\/latex]<\/td>\r\n<td>[latex]x=3\\text{\/}2[\/latex]<\/td>\r\n<td>[latex](\\text{+})(\\text{\u2212})\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](2,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=3[\/latex]<\/td>\r\n<td>[latex](\\text{+})(\\text{+})\\text{\/}+=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042476453\">From this table, we see that [latex]f[\/latex] has a local maximum at [latex]x=0[\/latex] and a local minimum at [latex]x=2.[\/latex] The value of [latex]f[\/latex] at the local maximum is [latex]f(0)=0[\/latex] and the value of [latex]f[\/latex] at the local minimum is [latex]f(2)=4.[\/latex] Therefore, [latex](0,0)[\/latex] and [latex](2,4)[\/latex] are important points on the graph.<\/p>\r\n<p id=\"fs-id1165042464546\">Step 6. Calculate the second derivative:<\/p>\r\n\r\n<div id=\"fs-id1165042464549\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill f\\text{\u2033}(x)&amp; =\\frac{{(x-1)}^{2}(2x-2)-({x}^{2}-2x)(2(x-1))}{{(x-1)}^{4}}\\hfill \\\\ &amp; =\\frac{(x-1)\\left[(x-1)(2x-2)-2({x}^{2}-2x)\\right]}{{(x-1)}^{4}}\\hfill \\\\ &amp; =\\frac{(x-1)(2x-2)-2({x}^{2}-2x)}{{(x-1)}^{3}}\\hfill \\\\ &amp; =\\frac{2{x}^{2}-4x+2-(2{x}^{2}-4x)}{{(x-1)}^{3}}\\hfill \\\\ &amp; =\\frac{2}{{(x-1)}^{3}}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165042652400\">We see that [latex]f\\text{\u2033}(x)[\/latex] is never zero or undefined for [latex]x[\/latex] in the domain of [latex]f.[\/latex] Since [latex]f[\/latex] is undefined at [latex]x=1,[\/latex] to check concavity we just divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the two smaller intervals [latex](\\text{\u2212}\\infty ,1)[\/latex] and [latex](1,\\infty ),[\/latex] and choose a test point from each interval to evaluate the sign of [latex]f\\text{\u2033}(x)[\/latex] in each of these intervals. The values [latex]x=0[\/latex] and [latex]x=2[\/latex] are possible test points as shown in the following table.<\/p>\r\n\r\n<table id=\"fs-id1165042652542\" class=\"unnumbered\" summary=\"This table has four columns and three rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019\u2019(x) = 2\/(x \u2212 1)3, and Conclusion. Under the header row, the first column reads (\u2212\u221e, 1) and (1, \u221e). The column row reads x = 0 and x = 2. The third column reads +\/\u2212 = \u2212 and +\/+ = +. The fourth column reads f is concave down and f is concave up.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]f\\text{\u2033}(x)=\\frac{2}{{(x-1)}^{3}}[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\\text{\u2212}\\infty ,1)[\/latex]<\/td>\r\n<td>[latex]x=0[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}-=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave down.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](1,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=2[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave up.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042606958\">From the information gathered, we arrive at the following graph for [latex]f.[\/latex]<\/p>\r\n<span id=\"fs-id1165042606968\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211145\/CNX_Calc_Figure_04_06_017.jpg\" alt=\"The function f(x) = x2\/(x \u2212 1) is graphed. It has asymptotes y = x + 1 and x = 1.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042606982\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042606986\" class=\"exercise\">\r\n<div id=\"fs-id1165042606988\" class=\"textbox\">\r\n<p id=\"fs-id1165042606991\">Find the oblique asymptote for [latex]f(x)=\\frac{(3{x}^{3}-2x+1)}{(2{x}^{2}-4)}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042607059\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042607059\"]\r\n<p id=\"fs-id1165042607059\">[latex]y=\\frac{3}{2}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042607076\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042607084\">Use long division of polynomials.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042607090\" class=\"textbox examples\">\r\n<h3>Sketching the Graph of a Function with a Cusp<\/h3>\r\n<div id=\"fs-id1165042607093\" class=\"exercise\">\r\n<div id=\"fs-id1165042607095\" class=\"textbox\">\r\n<p id=\"fs-id1165042607100\">Sketch a graph of [latex]f(x)={(x-1)}^{2\\text{\/}3}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042607145\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042607145\"]\r\n<p id=\"fs-id1165042607145\">Step 1. Since the cube-root function is defined for all real numbers [latex]x[\/latex] and [latex]{(x-1)}^{2\\text{\/}3}={(\\sqrt[3]{x-1})}^{2},[\/latex] the domain of [latex]f[\/latex] is all real numbers.<\/p>\r\n<p id=\"fs-id1165042607208\">Step 2: To find the [latex]y[\/latex]-intercept, evaluate [latex]f(0).[\/latex] Since [latex]f(0)=1,[\/latex] the [latex]y[\/latex]-intercept is [latex](0,1).[\/latex] To find the [latex]x[\/latex]-intercept, solve [latex]{(x-1)}^{2\\text{\/}3}=0.[\/latex] The solution of this equation is [latex]x=1,[\/latex] so the [latex]x[\/latex]-intercept is [latex](1,0).[\/latex]<\/p>\r\n<p id=\"fs-id1165042583208\">Step 3: Since [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}{(x-1)}^{2\\text{\/}3}=\\infty ,[\/latex] the function continues to grow without bound as [latex]x\\to \\infty [\/latex] and [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\r\n<p id=\"fs-id1165042583285\">Step 4: The function has no vertical asymptotes.<\/p>\r\n<p id=\"fs-id1165042583288\">Step 5: To determine where [latex]f[\/latex] is increasing or decreasing, calculate [latex]{f}^{\\prime }.[\/latex] We find<\/p>\r\n\r\n<div id=\"fs-id1165042583307\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{2}{3}{(x-1)}^{-1\\text{\/}3}=\\frac{2}{3{(x-1)}^{1\\text{\/}3}}.[\/latex]<\/div>\r\n<p id=\"fs-id1165042583388\">This function is not zero anywhere, but it is undefined when [latex]x=1.[\/latex] Therefore, the only critical point is [latex]x=1.[\/latex] Divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the smaller intervals [latex](\\text{\u2212}\\infty ,1)[\/latex] and [latex](1,\\infty ),[\/latex] and choose test points in each of these intervals to determine the sign of [latex]{f}^{\\prime }(x)[\/latex] in each of these smaller intervals. Let [latex]x=0[\/latex] and [latex]x=2[\/latex] be the test points as shown in the following table.<\/p>\r\n\r\n<table id=\"fs-id1165042504467\" class=\"unnumbered\" summary=\"This table has four columns and three rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019(x) = 2\/(3(x \u2212 1)1\/3), and Conclusion. Under the header row, the first column reads (\u2212\u221e, 1) and (1, \u221e). The second column reads x = 0 and x = 2. The third column reads +\/\u2212 = \u2212 and +\/+ = +. The fourth column reads f is decreasing and f is increasing.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]{f}^{\\prime }(x)=\\frac{2}{3{(x-1)}^{1\\text{\/}3}}[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\\text{\u2212}\\infty ,1)[\/latex]<\/td>\r\n<td>[latex]x=0[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}-=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](1,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=2[\/latex]<\/td>\r\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042504734\">We conclude that [latex]f[\/latex] has a local minimum at [latex]x=1.[\/latex] Evaluating [latex]f[\/latex] at [latex]x=1,[\/latex] we find that the value of [latex]f[\/latex] at the local minimum is zero. Note that [latex]{f}^{\\prime }(1)[\/latex] is undefined, so to determine the behavior of the function at this critical point, we need to examine [latex]\\underset{x\\to 1}{\\text{lim}}{f}^{\\prime }(x).[\/latex] Looking at the one-sided limits, we have<\/p>\r\n\r\n<div id=\"fs-id1165042510168\" class=\"equation unnumbered\">[latex]\\underset{x\\to {1}^{+}}{\\text{lim}}\\frac{2}{3{(x-1)}^{1\\text{\/}3}}=\\infty \\text{ and }\\underset{x\\to {1}^{-}}{\\text{lim}}\\frac{2}{3{(x-1)}^{1\\text{\/}3}}=\\text{\u2212}\\infty .[\/latex]<\/div>\r\n<p id=\"fs-id1165042510285\">Therefore, [latex]f[\/latex] has a cusp at [latex]x=1.[\/latex]<\/p>\r\n<p id=\"fs-id1165042510304\">Step 6: To determine concavity, we calculate the second derivative of [latex]f\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165042510313\" class=\"equation unnumbered\">[latex]f\\text{\u2033}(x)=-\\frac{2}{9}{(x-1)}^{-4\\text{\/}3}=\\frac{-2}{9{(x-1)}^{4\\text{\/}3}}.[\/latex]<\/div>\r\n<p id=\"fs-id1165042510398\">We find that [latex]f\\text{\u2033}(x)[\/latex] is defined for all [latex]x,[\/latex] but is undefined when [latex]x=1.[\/latex] Therefore, divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the smaller intervals [latex](\\text{\u2212}\\infty ,1)[\/latex] and [latex](1,\\infty ),[\/latex] and choose test points to evaluate the sign of [latex]f\\text{\u2033}(x)[\/latex] in each of these intervals. As we did earlier, let [latex]x=0[\/latex] and [latex]x=2[\/latex] be test points as shown in the following table.<\/p>\r\n\r\n<table id=\"fs-id1165042510530\" class=\"unnumbered\" summary=\"This table has four columns and three rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019\u2019(x) = \u22122\/(9(x \u2212 1)4\/3), and Conclusion. Under the header row, the first column reads (\u2212\u221e, 1) and (1, \u221e). The second column reads x = 0 and x = 2. The third column reads \u2212\/+ = \u2212 and \u2212\/+ = \u2212. The fourth column reads f is concave down and f is concave down.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]f\\text{\u2033}(x)=\\frac{-2}{9{(x-1)}^{4\\text{\/}3}}[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex](\\text{\u2212}\\infty ,1)[\/latex]<\/td>\r\n<td>[latex]x=0[\/latex]<\/td>\r\n<td>[latex]\\text{\u2212}\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave down.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](1,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=2[\/latex]<\/td>\r\n<td>[latex]\\text{\u2212}\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is concave down.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042643969\">From this table, we conclude that [latex]f[\/latex] is concave down everywhere. Combining all of this information, we arrive at the following graph for [latex]f.[\/latex]<\/p>\r\n<span id=\"fs-id1165042643991\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211148\/CNX_Calc_Figure_04_06_018.jpg\" alt=\"The function f(x) = (x \u2212 1)2\/3 is graphed. It touches the x axis at x = 1, where it comes to something of a sharp point and then flairs out on either side.\" \/><\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042644007\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042644011\" class=\"exercise\">\r\n<div id=\"fs-id1165042644013\" class=\"textbox\">\r\n<p id=\"fs-id1165042644015\">Consider the function [latex]f(x)=5-{x}^{2\\text{\/}3}.[\/latex] Determine the point on the graph where a cusp is located. Determine the end behavior of [latex]f.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042644060\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042644060\"]\r\n<p id=\"fs-id1165042644060\">The function [latex]f[\/latex] has a cusp at [latex](0,5)[\/latex] [latex]\\underset{x\\to {0}^{-}}{\\text{lim}}{f}^{\\prime }(x)=\\infty ,[\/latex] [latex]\\underset{x\\to {0}^{+}}{\\text{lim}}{f}^{\\prime }(x)=\\text{\u2212}\\infty .[\/latex] For end behavior, [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty .[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042633566\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042633573\">A function [latex]f[\/latex] has a cusp at a point [latex]a[\/latex] if [latex]f(a)[\/latex] exists, [latex]f\\prime (a)[\/latex] is undefined, one of the one-sided limits as [latex]x\\to a[\/latex] of [latex]f\\prime (x)[\/latex] is [latex]+\\infty ,[\/latex] and the other one-sided limit is [latex]\\text{\u2212}\\infty .[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042633662\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165042633670\">\r\n \t<li>The limit of [latex]f(x)[\/latex] is [latex]L[\/latex] as [latex]x\\to \\infty [\/latex] (or as [latex]x\\to \\text{\u2212}\\infty )[\/latex] if the values [latex]f(x)[\/latex] become arbitrarily close to [latex]L[\/latex] as [latex]x[\/latex] becomes sufficiently large.<\/li>\r\n \t<li>The limit of [latex]f(x)[\/latex] is [latex]\\infty [\/latex] as [latex]x\\to \\infty [\/latex] if [latex]f(x)[\/latex] becomes arbitrarily large as [latex]x[\/latex] becomes sufficiently large. The limit of [latex]f(x)[\/latex] is [latex]\\text{\u2212}\\infty [\/latex] as [latex]x\\to \\infty [\/latex] if [latex]f(x)&lt;0[\/latex] and [latex]|f(x)|[\/latex] becomes arbitrarily large as [latex]x[\/latex] becomes sufficiently large. We can define the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]\\text{\u2212}\\infty [\/latex] similarly.<\/li>\r\n \t<li>For a polynomial function [latex]p(x)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\\text{\u2026}+{a}_{1}x+{a}_{0},[\/latex] where [latex]{a}_{n}\\ne 0,[\/latex] the end behavior is determined by the leading term [latex]{a}_{n}{x}^{n}.[\/latex] If [latex]n\\ne 0,[\/latex] [latex]p(x)[\/latex] approaches [latex]\\infty [\/latex] or [latex]\\text{\u2212}\\infty [\/latex] at each end.<\/li>\r\n \t<li>For a rational function [latex]f(x)=\\frac{p(x)}{q(x)},[\/latex] the end behavior is determined by the relationship between the degree of [latex]p[\/latex] and the degree of [latex]q.[\/latex] If the degree of [latex]p[\/latex] is less than the degree of [latex]q,[\/latex] the line [latex]y=0[\/latex] is a horizontal asymptote for [latex]f.[\/latex] If the degree of [latex]p[\/latex] is equal to the degree of [latex]q,[\/latex] then the line [latex]y=\\frac{{a}_{n}}{{b}_{n}}[\/latex] is a horizontal asymptote, where [latex]{a}_{n}[\/latex] and [latex]{b}_{n}[\/latex] are the leading coefficients of [latex]p[\/latex] and [latex]q,[\/latex] respectively. If the degree of [latex]p[\/latex] is greater than the degree of [latex]q,[\/latex] then [latex]f[\/latex] approaches [latex]\\infty [\/latex] or [latex]\\text{\u2212}\\infty [\/latex] at each end.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165042640624\" class=\"textbox exercises\">\r\n<p id=\"fs-id1165042640628\">For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.<\/p>\r\n\r\n<div id=\"fs-id1165042640632\" class=\"exercise\">\r\n<div id=\"fs-id1165042640635\" class=\"textbox\"><span id=\"fs-id1165042640637\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211151\/CNX_Calc_Figure_04_06_201.jpg\" alt=\"The function graphed decreases very rapidly as it approaches x = 1 from the left, and on the other side of x = 1, it seems to start near infinity and then decrease rapidly.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042640653\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042640653\"]\r\n<p id=\"fs-id1165042640653\">[latex]x=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042640666\" class=\"exercise\">\r\n<div id=\"fs-id1165042640668\" class=\"textbox\"><span id=\"fs-id1165042640674\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211153\/CNX_Calc_Figure_04_06_202.jpg\" alt=\"The function graphed increases very rapidly as it approaches x = \u22123 from the left, and on the other side of x = \u22123, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043197220\" class=\"exercise\">\r\n<div id=\"fs-id1165043197222\" class=\"textbox\"><span id=\"fs-id1165043197228\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211156\/CNX_Calc_Figure_04_06_203.jpg\" alt=\"The function graphed decreases very rapidly as it approaches x = \u22121 from the left, and on the other side of x = \u22121, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043197243\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043197243\"]\r\n<p id=\"fs-id1165043197243\">[latex]x=-1,x=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043197265\" class=\"exercise\">\r\n<div id=\"fs-id1165043197267\" class=\"textbox\"><span id=\"fs-id1165043197269\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211158\/CNX_Calc_Figure_04_06_204.jpg\" alt=\"The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side of the U being at x = 1. On the other side of x = 1, there is another U shape pointing down, with its other side being at x = 2. On the other side of x = 2, the graph seems to start near negative infinity and then increase rapidly.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043197318\" class=\"exercise\">\r\n<div id=\"fs-id1165043197320\" class=\"textbox\"><span id=\"fs-id1165043197322\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211201\/CNX_Calc_Figure_04_06_205.jpg\" alt=\"The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side being a normal function that appears as if it will take the entirety of the values of the x-axis.\" \/><\/span><\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043197340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043197340\"]\r\n<p id=\"fs-id1165043197340\">[latex]x=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043197353\">For the following functions [latex]f(x),[\/latex] determine whether there is an asymptote at [latex]x=a.[\/latex] Justify your answer without graphing on a calculator.<\/p>\r\n\r\n<div id=\"fs-id1165043197383\" class=\"exercise\">\r\n<div id=\"fs-id1165043197385\" class=\"textbox\">\r\n<p id=\"fs-id1165043197388\">[latex]f(x)=\\frac{x+1}{{x}^{2}+5x+4},a=-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043197447\" class=\"exercise\">\r\n<div id=\"fs-id1165043197449\" class=\"textbox\">\r\n<p id=\"fs-id1165043197451\">[latex]f(x)=\\frac{x}{x-2},a=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043197490\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043197490\"]\r\n<p id=\"fs-id1165043197490\">Yes, there is a vertical asymptote<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043197495\" class=\"exercise\">\r\n<div id=\"fs-id1165043197497\" class=\"textbox\">\r\n<p id=\"fs-id1165043197500\">[latex]f(x)={(x+2)}^{3\\text{\/}2},a=-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043197555\" class=\"exercise\">\r\n<div id=\"fs-id1165043197557\" class=\"textbox\">\r\n<p id=\"fs-id1165043197560\">[latex]f(x)={(x-1)}^{-1\\text{\/}3},a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043197610\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043197610\"]\r\n<p id=\"fs-id1165043197610\">Yes, there is vertical asymptote<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043197616\" class=\"exercise\">\r\n<div id=\"fs-id1165043197618\" class=\"textbox\">\r\n<p id=\"fs-id1165043197620\">[latex]f(x)=1+{x}^{-2\\text{\/}5},a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043377731\">For the following exercises, evaluate the limit.<\/p>\r\n\r\n<div id=\"fs-id1165043377734\" class=\"exercise\">\r\n<div id=\"fs-id1165043377736\" class=\"textbox\">\r\n<p id=\"fs-id1165043377738\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{3x+6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043377772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043377772\"]\r\n<p id=\"fs-id1165043377772\">0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043377780\" class=\"exercise\">\r\n<div id=\"fs-id1165043377782\" class=\"textbox\">\r\n<p id=\"fs-id1165043377784\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{2x-5}{4x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043377833\" class=\"exercise\">\r\n<div id=\"fs-id1165043377835\" class=\"textbox\">\r\n<p id=\"fs-id1165043377837\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{{x}^{2}-2x+5}{x+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043377884\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043377884\"]\r\n<p id=\"fs-id1165043377884\">[latex]\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043377891\" class=\"exercise\">\r\n<div id=\"fs-id1165043377893\" class=\"textbox\">\r\n<p id=\"fs-id1165043377896\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{3{x}^{3}-2x}{{x}^{2}+2x+8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043377963\" class=\"exercise\">\r\n<div id=\"fs-id1165043377965\" class=\"textbox\">\r\n<p id=\"fs-id1165043377967\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{{x}^{4}-4{x}^{3}+1}{2-2{x}^{2}-7{x}^{4}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043378034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043378034\"]\r\n<p id=\"fs-id1165043378034\">[latex]-\\frac{1}{7}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043378048\" class=\"exercise\">\r\n<div id=\"fs-id1165043378050\" class=\"textbox\">\r\n<p id=\"fs-id1165043378052\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{3x}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043378100\" class=\"exercise\">\r\n<div id=\"fs-id1165043216345\" class=\"textbox\">\r\n<p id=\"fs-id1165043216347\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{\\sqrt{4{x}^{2}-1}}{x+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043216394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043216394\"]\r\n<p id=\"fs-id1165043216394\">-2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043216402\" class=\"exercise\">\r\n<div id=\"fs-id1165043216404\" class=\"textbox\">\r\n<p id=\"fs-id1165043216407\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{4x}{\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043216454\" class=\"exercise\">\r\n<div id=\"fs-id1165043216457\" class=\"textbox\">\r\n<p id=\"fs-id1165043216459\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{4x}{\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043216501\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043216501\"]\r\n<p id=\"fs-id1165043216501\">-4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043216510\" class=\"exercise\">\r\n<div id=\"fs-id1165043216512\" class=\"textbox\">\r\n<p id=\"fs-id1165043216514\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{2\\sqrt{x}}{x-\\sqrt{x}+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043216563\">For the following exercises, find the horizontal and vertical asymptotes.<\/p>\r\n\r\n<div id=\"fs-id1165043216566\" class=\"exercise\">\r\n<div id=\"fs-id1165043216569\" class=\"textbox\">\r\n<p id=\"fs-id1165043216571\">[latex]f(x)=x-\\frac{9}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043216600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043216600\"]\r\n<p id=\"fs-id1165043216600\">Horizontal: none, vertical: [latex]x=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043216614\" class=\"exercise\">\r\n<div id=\"fs-id1165043216616\" class=\"textbox\">\r\n<p id=\"fs-id1165043216618\">[latex]f(x)=\\frac{1}{1-{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043216679\" class=\"exercise\">\r\n<div id=\"fs-id1165043216681\" class=\"textbox\">\r\n<p id=\"fs-id1165043216683\">[latex]f(x)=\\frac{{x}^{3}}{4-{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043216720\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043216720\"]\r\n<p id=\"fs-id1165043216720\">Horizontal: none, vertical: [latex]x=\\text{\u00b1}2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043216737\" class=\"exercise\">\r\n<div id=\"fs-id1165043216739\" class=\"textbox\">\r\n<p id=\"fs-id1165043216741\">[latex]f(x)=\\frac{{x}^{2}+3}{{x}^{2}+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043210294\" class=\"exercise\">\r\n<div id=\"fs-id1165043210296\" class=\"textbox\">\r\n<p id=\"fs-id1165043210298\">[latex]f(x)= \\sin (x) \\sin (2x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043210340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043210340\"]\r\n<p id=\"fs-id1165043210340\">Horizontal: none, vertical: none<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043210346\" class=\"exercise\">\r\n<div id=\"fs-id1165043210348\" class=\"textbox\">\r\n<p id=\"fs-id1165043210350\">[latex]f(x)= \\cos x+ \\cos (3x)+ \\cos (5x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043210409\" class=\"exercise\">\r\n<div id=\"fs-id1165043210411\" class=\"textbox\">\r\n<p id=\"fs-id1165043210413\">[latex]f(x)=\\frac{x \\sin (x)}{{x}^{2}-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043210457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043210457\"]\r\n<p id=\"fs-id1165043210457\">Horizontal: [latex]y=0,[\/latex] vertical: [latex]x=\\text{\u00b1}1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043210485\" class=\"exercise\">\r\n<div id=\"fs-id1165043210487\" class=\"textbox\">\r\n<p id=\"fs-id1165043210489\">[latex]f(x)=\\frac{x}{ \\sin (x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043210550\" class=\"exercise\">\r\n<div id=\"fs-id1165043210552\" class=\"textbox\">\r\n<p id=\"fs-id1165043210554\">[latex]f(x)=\\frac{1}{{x}^{3}+{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043210591\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043210591\"]\r\n<p id=\"fs-id1165043210591\">Horizontal: [latex]y=0,[\/latex] vertical: [latex]x=0[\/latex] and [latex]x=-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043210626\" class=\"exercise\">\r\n<div id=\"fs-id1165043210628\" class=\"textbox\">\r\n<p id=\"fs-id1165043210630\">[latex]f(x)=\\frac{1}{x-1}-2x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043349198\" class=\"exercise\">\r\n<div id=\"fs-id1165043349200\" class=\"textbox\">\r\n<p id=\"fs-id1165043349202\">[latex]f(x)=\\frac{{x}^{3}+1}{{x}^{3}-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043349244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043349244\"]\r\n<p id=\"fs-id1165043349244\">Horizontal: [latex]y=1,[\/latex] vertical: [latex]x=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043349269\" class=\"exercise\">\r\n<div id=\"fs-id1165043349272\" class=\"textbox\">\r\n<p id=\"fs-id1165043349274\">[latex]f(x)=\\frac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043349346\" class=\"exercise\">\r\n<div id=\"fs-id1165043349348\" class=\"textbox\">\r\n<p id=\"fs-id1165043349350\">[latex]f(x)=x- \\sin x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043349378\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043349378\"]\r\n<p id=\"fs-id1165043349378\">Horizontal: none, vertical: none<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043349383\" class=\"exercise\">\r\n<div id=\"fs-id1165043349385\" class=\"textbox\">\r\n<p id=\"fs-id1165043349388\">[latex]f(x)=\\frac{1}{x}-\\sqrt{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043349432\">For the following exercises, construct a function [latex]f(x)[\/latex] that has the given asymptotes.<\/p>\r\n\r\n<div id=\"fs-id1165043349448\" class=\"exercise\">\r\n<div id=\"fs-id1165043349450\" class=\"textbox\">\r\n<p id=\"fs-id1165043349452\">[latex]x=1[\/latex] and [latex]y=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043349476\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043349476\"]\r\n<p id=\"fs-id1165043349476\">Answers will vary, for example: [latex]y=\\frac{2x}{x-1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043349502\" class=\"exercise\">\r\n<div id=\"fs-id1165043349504\" class=\"textbox\">\r\n<p id=\"fs-id1165043349506\">[latex]x=1[\/latex] and [latex]y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043349559\" class=\"exercise\">\r\n<div id=\"fs-id1165043349561\" class=\"textbox\">\r\n<p id=\"fs-id1165043349563\">[latex]y=4,[\/latex][latex]x=-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043349589\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043349589\"]\r\n<p id=\"fs-id1165043349589\">Answers will vary, for example: [latex]y=\\frac{4x}{x+1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043349615\" class=\"exercise\">\r\n<div id=\"fs-id1165043349617\" class=\"textbox\">\r\n<p id=\"fs-id1165043349619\">[latex]x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042709113\">For the following exercises, graph the function on a graphing calculator on the window [latex]x=\\left[-5,5\\right][\/latex] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.<\/p>\r\n\r\n<div id=\"fs-id1165042709138\" class=\"exercise\">\r\n<div id=\"fs-id1165042709140\" class=\"textbox\">\r\n<p id=\"fs-id1165042709142\"><strong>[T]<\/strong>[latex]f(x)=\\frac{1}{x+10}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042709177\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042709177\"]\r\n<p id=\"fs-id1165042709177\">[latex]y=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042709190\" class=\"exercise\">\r\n<div id=\"fs-id1165042709192\" class=\"textbox\">\r\n<p id=\"fs-id1165042709194\"><strong>[T]<\/strong>[latex]f(x)=\\frac{x+1}{{x}^{2}+7x+6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042709257\" class=\"exercise\">\r\n<div id=\"fs-id1165042709259\" class=\"textbox\">\r\n<p id=\"fs-id1165042709261\"><strong>[T]<\/strong>[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{2}+10x+25[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042709306\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042709306\"]\r\n<p id=\"fs-id1165042709306\">[latex]\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042709313\" class=\"exercise\">\r\n<div id=\"fs-id1165042709316\" class=\"textbox\">\r\n<p id=\"fs-id1165042709318\"><strong>[T]<\/strong>[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{x+2}{{x}^{2}+7x+6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042709385\" class=\"exercise\">\r\n<div id=\"fs-id1165042709387\" class=\"textbox\">\r\n<p id=\"fs-id1165042709389\"><strong>[T]<\/strong>[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{3x+2}{x+5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042709433\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042709433\"]\r\n<p id=\"fs-id1165042709433\">[latex]y=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042709446\">For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.<\/p>\r\n\r\n<div id=\"fs-id1165042709451\" class=\"exercise\">\r\n<div id=\"fs-id1165042709453\" class=\"textbox\">\r\n<p id=\"fs-id1165042709456\">[latex]y=3{x}^{2}+2x+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042709501\" class=\"exercise\">\r\n<div id=\"fs-id1165042709504\" class=\"textbox\">\r\n<p id=\"fs-id1165042709506\">[latex]y={x}^{3}-3{x}^{2}+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042558120\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042558120\"]<span id=\"fs-id1165042558126\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211203\/CNX_Calc_Figure_04_06_207.jpg\" alt=\"The function starts in the third quadrant, increases to pass through (\u22121, 0), increases to a maximum and y intercept at 4, decreases to touch (2, 0), and then increases to (4, 20).\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042558140\" class=\"exercise\">\r\n<div id=\"fs-id1165042558142\" class=\"textbox\">\r\n<p id=\"fs-id1165042558144\">[latex]y=\\frac{2x+1}{{x}^{2}+6x+5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042558205\" class=\"exercise\">\r\n<div id=\"fs-id1165042558207\" class=\"textbox\">\r\n<p id=\"fs-id1165042558209\">[latex]y=\\frac{{x}^{3}+4{x}^{2}+3x}{3x+9}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042558253\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042558253\"]<span id=\"fs-id1165042558258\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211205\/CNX_Calc_Figure_04_06_209.jpg\" alt=\"An upward-facing parabola with minimum between x = 0 and x = \u22121 with y intercept between 0 and 1.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042558272\" class=\"exercise\">\r\n<div id=\"fs-id1165042558274\" class=\"textbox\">\r\n<p id=\"fs-id1165042558276\">[latex]y=\\frac{{x}^{2}+x-2}{{x}^{2}-3x-4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042558343\" class=\"exercise\">\r\n<div id=\"fs-id1165042558345\" class=\"textbox\">\r\n<p id=\"fs-id1165042558347\">[latex]y=\\sqrt{{x}^{2}-5x+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042558376\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042558376\"]<span id=\"fs-id1165042558381\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211208\/CNX_Calc_Figure_04_06_211.jpg\" alt=\"This graph starts at (\u22122, 4) and decreases in a convex way to (1, 0). Then the graph starts again at (4, 0) and increases in a convex way to (6, 3).\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042558395\" class=\"exercise\">\r\n<div id=\"fs-id1165042558397\" class=\"textbox\">\r\n<p id=\"fs-id1165042558399\">[latex]y=2x\\sqrt{16-{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042558446\" class=\"exercise\">\r\n<div id=\"fs-id1165042558448\" class=\"textbox\">\r\n<p id=\"fs-id1165042558450\">[latex]y=\\frac{ \\cos x}{x},[\/latex] on [latex]x=\\left[-2\\pi ,2\\pi \\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042558496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042558496\"]<span id=\"fs-id1165042558507\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211210\/CNX_Calc_Figure_04_06_213.jpg\" alt=\"This graph has vertical asymptote at x = 0. The first part of the function occurs in the second and third quadrants and starts in the third quadrant just below (\u22122\u03c0, 0), increases and passes through the x axis at \u22123\u03c0\/2, reaches a maximum and then decreases through the x axis at \u2212\u03c0\/2 before approaching the asymptote. On the other side of the asymptote, the function starts in the first quadrant, decreases quickly to pass through \u03c0\/2, decreases to a local minimum and then increases through (3\u03c0\/2, 0) before staying just above (2\u03c0, 0).\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042558519\" class=\"exercise\">\r\n<div id=\"fs-id1165042558521\" class=\"textbox\">\r\n<p id=\"fs-id1165042558523\">[latex]y={e}^{x}-{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043404024\" class=\"exercise\">\r\n<div id=\"fs-id1165043404026\" class=\"textbox\">\r\n<p id=\"fs-id1165043404029\">[latex]y=x \\tan x,x=\\left[\\text{\u2212}\\pi ,\\pi \\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043404067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043404067\"]<span id=\"fs-id1165043404074\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211213\/CNX_Calc_Figure_04_06_215.jpg\" alt=\"This graph has vertical asymptotes at x = \u00b1\u03c0\/2. The graph is symmetric about the y axis, so describing the left hand side will be sufficient. The function starts at (\u2212\u03c0, 0) and decreases quickly to the asymptote. Then it starts on the other side of the asymptote in the second quadrant and decreases to the the origin.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043404088\" class=\"exercise\">\r\n<div id=\"fs-id1165043404090\" class=\"textbox\">\r\n<p id=\"fs-id1165043404092\">[latex]y=x\\text{ln}(x),x&gt;0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043404141\" class=\"exercise\">\r\n<div id=\"fs-id1165043404143\" class=\"textbox\">\r\n<p id=\"fs-id1165043404145\">[latex]y={x}^{2} \\sin (x),x=\\left[-2\\pi ,2\\pi \\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043404194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043404194\"]<span id=\"fs-id1165043404201\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211215\/CNX_Calc_Figure_04_06_217.jpg\" alt=\"This function starts at (\u22122\u03c0, 0), increases to near (\u22123\u03c0\/2, 25), decreases through (\u2212\u03c0, 0), achieves a local minimum and then increases through the origin. On the other side of the origin, the graph is the same but flipped, that is, it is congruent to the other half by a rotation of 180 degrees.\" \/><\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043404215\" class=\"exercise\">\r\n<div id=\"fs-id1165043404217\" class=\"textbox\">\r\n<p id=\"fs-id1165043404219\">For [latex]f(x)=\\frac{P(x)}{Q(x)}[\/latex] to have an asymptote at [latex]y=2[\/latex] then the polynomials [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] must have what relation?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043404361\" class=\"exercise\">\r\n<div id=\"fs-id1165043404363\" class=\"textbox\">\r\n<p id=\"fs-id1165043404366\">For [latex]f(x)=\\frac{P(x)}{Q(x)}[\/latex] to have an asymptote at [latex]x=0,[\/latex] then the polynomials [latex]P(x)[\/latex] and [latex]Q(x).[\/latex] must have what relation?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043208543\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043208543\"]\r\n<p id=\"fs-id1165043208543\">[latex]Q(x).[\/latex] must have have [latex]{x}^{k+1}[\/latex] as a factor, where [latex]P(x)[\/latex] has [latex]{x}^{k}[\/latex] as a factor.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043208598\" class=\"exercise\">\r\n<div id=\"fs-id1165043208600\" class=\"textbox\">\r\n<p id=\"fs-id1165043208602\">If [latex]{f}^{\\prime }(x)[\/latex] has asymptotes at [latex]y=3[\/latex] and [latex]x=1,[\/latex] then [latex]f(x)[\/latex] has what asymptotes?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043208676\" class=\"exercise\">\r\n<div id=\"fs-id1165043208678\" class=\"textbox\">\r\n<p id=\"fs-id1165043208681\">Both [latex]f(x)=\\frac{1}{(x-1)}[\/latex] and [latex]g(x)=\\frac{1}{{(x-1)}^{2}}[\/latex] have asymptotes at [latex]x=1[\/latex] and [latex]y=0.[\/latex] What is the most obvious difference between these two functions?<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043208777\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043208777\"]\r\n<p id=\"fs-id1165043208777\">[latex]\\underset{x\\to {1}^{-}}{\\text{lim}}f(x)\\text{ and }\\underset{x\\to {1}^{-}}{\\text{lim}}g(x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043208845\" class=\"exercise\">\r\n<div id=\"fs-id1165043208847\" class=\"textbox\">\r\n<p id=\"fs-id1165043208849\">True or false: Every ratio of polynomials has vertical asymptotes.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165043208865\" class=\"definition\">\r\n \t<dt>end behavior<\/dt>\r\n \t<dd id=\"fs-id1165043208870\">the behavior of a function as [latex]x\\to \\infty [\/latex] and [latex]x\\to \\text{\u2212}\\infty [\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043208899\" class=\"definition\">\r\n \t<dt>horizontal asymptote<\/dt>\r\n \t<dd id=\"fs-id1165043208905\">if [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L[\/latex] or [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=L,[\/latex] then [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042462524\" class=\"definition\">\r\n \t<dt>infinite limit at infinity<\/dt>\r\n \t<dd id=\"fs-id1165042462530\">a function that becomes arbitrarily large as [latex]x[\/latex] becomes large<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042462539\" class=\"definition\">\r\n \t<dt>limit at infinity<\/dt>\r\n \t<dd id=\"fs-id1165042462545\">the limiting value, if it exists, of a function as [latex]x\\to \\infty [\/latex] or [latex]x\\to \\text{\u2212}\\infty [\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042462574\" class=\"definition\">\r\n \t<dt>oblique asymptote<\/dt>\r\n \t<dd id=\"fs-id1165042462579\">the line [latex]y=mx+b[\/latex] if [latex]f(x)[\/latex] approaches it as [latex]x\\to \\infty [\/latex] or [latex]x\\to \\text{\u2212}\\infty [\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Calculate the limit of a function as [latex]x[\/latex] increases or decreases without bound.<\/li>\n<li>Recognize a horizontal asymptote on the graph of a function.<\/li>\n<li>Estimate the end behavior of a function as [latex]x[\/latex] increases or decreases without bound.<\/li>\n<li>Recognize an oblique asymptote on the graph of a function.<\/li>\n<li>Analyze a function and its derivatives to draw its graph.<\/li>\n<\/ul>\n<\/div>\n<p>We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function [latex]f[\/latex] defined on an unbounded domain, we also need to know the behavior of [latex]f[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function [latex]f.[\/latex]<\/p>\n<div id=\"fs-id1165043145047\" class=\"bc-section section\">\n<h1>Limits at Infinity<\/h1>\n<p id=\"fs-id1165043308392\">We begin by examining what it means for a function to have a finite <strong>limit at infinity.<\/strong> Then we study the idea of a function with an <strong>infinite limit at infinity<\/strong>. Back in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction\/\">Introduction to Functions and Graphs<\/a>, we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.<\/p>\n<div id=\"fs-id1165042704801\" class=\"bc-section section\">\n<h2>Limits at Infinity and Horizontal Asymptotes<\/h2>\n<p id=\"fs-id1165043107285\">Recall that [latex]\\underset{x\\to a}{\\text{lim}}f(x)=L[\/latex] means [latex]f(x)[\/latex] becomes arbitrarily close to [latex]L[\/latex] as long as [latex]x[\/latex] is sufficiently close to [latex]a.[\/latex] We can extend this idea to limits at infinity. For example, consider the function [latex]f(x)=2+\\frac{1}{x}.[\/latex] As can be seen graphically in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_001\">(Figure)<\/a> and numerically in <a class=\"autogenerated-content\" href=\"#fs-id1165043428402\">(Figure)<\/a>, as the values of [latex]x[\/latex] get larger, the values of [latex]f(x)[\/latex] approach 2. We say the limit as [latex]x[\/latex] approaches [latex]\\infty[\/latex] of [latex]f(x)[\/latex] is 2 and write [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=2.[\/latex] Similarly, for [latex]x<0,[\/latex] as the values [latex]|x|[\/latex] get larger, the values of [latex]f(x)[\/latex] approaches 2. We say the limit as [latex]x[\/latex] approaches [latex]\\text{\u2212}\\infty[\/latex] of [latex]f(x)[\/latex] is 2 and write [latex]\\underset{x\\to a}{\\text{lim}}f(x)=2.[\/latex]<\/p>\n<div id=\"CNX_Calc_Figure_04_06_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 727px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211025\/CNX_Calc_Figure_04_06_019.jpg\" alt=\"The function f(x) 2 + 1\/x is graphed. The function starts negative near y = 2 but then decreases to \u2212\u221e near x = 0. The function then decreases from \u221e near x = 0 and gets nearer to y = 2 as x increases. There is a horizontal line denoting the asymptote y = 2.\" width=\"717\" height=\"423\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.<\/strong> The function approaches the asymptote [latex]y=2[\/latex] as [latex]x[\/latex] approaches [latex]\\text{\u00b1}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<table id=\"fs-id1165043428402\" class=\"column-header\" summary=\"The table has four rows and five columns. The first column is a header column and it reads x, 2 + 1\/x, x, and 2 + 1\/x. After the header, the first row reads 10, 100, 1000, and 10000. The second row reads 2.1, 2.01, 2.001, and 2.0001. The third row reads \u221210, \u2212100, \u22121000, and \u221210000. The fourth row reads 1.9, 1.99, 1.999, and 1.9999.\">\n<caption>Values of a function [latex]f[\/latex] as [latex]x\\to \\text{\u00b1}\\infty[\/latex]<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>10<\/td>\n<td>100<\/td>\n<td>1,000<\/td>\n<td>10,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]2+\\frac{1}{x}[\/latex]<\/strong><\/td>\n<td>2.1<\/td>\n<td>2.01<\/td>\n<td>2.001<\/td>\n<td>2.0001<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>-10<\/td>\n<td>-100<\/td>\n<td>-1000<\/td>\n<td>-10,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]2+\\frac{1}{x}[\/latex]<\/strong><\/td>\n<td>1.9<\/td>\n<td>1.99<\/td>\n<td>1.999<\/td>\n<td>1.9999<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042936244\">More generally, for any function [latex]f,[\/latex] we say the limit as [latex]x\\to \\infty[\/latex] of [latex]f(x)[\/latex] is [latex]L[\/latex] if [latex]f(x)[\/latex] becomes arbitrarily close to [latex]L[\/latex] as long as [latex]x[\/latex] is sufficiently large. In that case, we write [latex]\\underset{x\\to a}{\\text{lim}}f(x)=L.[\/latex] Similarly, we say the limit as [latex]x\\to \\text{\u2212}\\infty[\/latex] of [latex]f(x)[\/latex] is [latex]L[\/latex] if [latex]f(x)[\/latex] becomes arbitrarily close to [latex]L[\/latex] as long as [latex]x<0[\/latex] and [latex]|x|[\/latex] is sufficiently large. In that case, we write [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=L.[\/latex] We now look at the definition of a function having a limit at infinity.<\/p>\n<div id=\"fs-id1165042331960\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1165042970725\">(Informal) If the values of [latex]f(x)[\/latex] become arbitrarily close to [latex]L[\/latex] as [latex]x[\/latex] becomes sufficiently large, we say the function [latex]f[\/latex] has a limit at infinity and write<\/p>\n<div id=\"fs-id1165042986551\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L.[\/latex]<\/div>\n<p id=\"fs-id1165042374662\">If the values of [latex]f(x)[\/latex] becomes arbitrarily close to [latex]L[\/latex] for [latex]x<0[\/latex] as [latex]|x|[\/latex] becomes sufficiently large, we say that the function [latex]f[\/latex] has a limit at negative infinity and write<\/p>\n<div id=\"fs-id1165043105208\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L.[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1165043157752\">If the values [latex]f(x)[\/latex] are getting arbitrarily close to some finite value [latex]L[\/latex] as [latex]x\\to \\infty[\/latex] or [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the graph of [latex]f[\/latex] approaches the line [latex]y=L.[\/latex] In that case, the line [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_002\">(Figure)<\/a>). For example, for the function [latex]f(x)=\\frac{1}{x},[\/latex] since [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=0,[\/latex] the line [latex]y=0[\/latex] is a horizontal asymptote of [latex]f(x)=\\frac{1}{x}.[\/latex]<\/p>\n<div id=\"fs-id1165043262534\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1165042973921\">If [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L[\/latex] or [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=L,[\/latex] we say the line [latex]y=L[\/latex] is a <strong>horizontal asymptote<\/strong> of [latex]f.[\/latex]<\/p>\n<\/div>\n<div id=\"CNX_Calc_Figure_04_06_002\" class=\"wp-caption aligncenter\">\n<div style=\"width: 776px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211028\/CNX_Calc_Figure_04_06_020.jpg\" alt=\"The figure is broken up into two figures labeled a and b. Figure a shows a function f(x) approaching but never touching a horizontal dashed line labeled L from above. Figure b shows a function f(x) approaching but never a horizontal dashed line labeled M from below.\" width=\"766\" height=\"273\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.<\/strong> (a) As [latex]x\\to \\infty ,[\/latex] the values of [latex]f[\/latex] are getting arbitrarily close to [latex]L.[\/latex] The line [latex]y=L[\/latex] is a horizontal asymptote of [latex]f.[\/latex] (b) As [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the values of [latex]f[\/latex] are getting arbitrarily close to [latex]M.[\/latex] The line [latex]y=M[\/latex] is a horizontal asymptote of [latex]f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042647732\">A function cannot cross a vertical asymptote because the graph must approach infinity (or [latex]\\text{\u2212}\\infty )[\/latex] from at least one direction as [latex]x[\/latex] approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function [latex]f(x)=\\frac{( \\cos x)}{x}+1[\/latex] shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_003\">(Figure)<\/a> intersects the horizontal asymptote [latex]y=1[\/latex] an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_003\" class=\"wp-caption aligncenter\">\n<div style=\"width: 539px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211031\/CNX_Calc_Figure_04_06_002.jpg\" alt=\"The function f(x) = (cos x)\/x + 1 is shown. It decreases from (0, \u221e) and then proceeds to oscillate around y = 1 with decreasing amplitude.\" width=\"529\" height=\"230\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong> The graph of [latex]f(x)=( \\cos x)\\text{\/}x+1[\/latex] crosses its horizontal asymptote [latex]y=1[\/latex] an infinite number of times.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042373486\">The algebraic limit laws and squeeze theorem we introduced in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction-2\/\">Introduction to Limits<\/a> also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.<\/p>\n<div class=\"textbox examples\">\n<div class=\"exercise\">\n<div id=\"fs-id1165043429468\" class=\"textbox\">\n<h3>Computing Limits at Infinity<\/h3>\n<p id=\"fs-id1165043262623\">For each of the following functions [latex]f,[\/latex] evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x).[\/latex] Determine the horizontal asymptote(s) for [latex]f.[\/latex]<\/p>\n<ol id=\"fs-id1165042356111\" style=\"list-style-type: lower-alpha\">\n<li>[latex]f(x)=5-\\frac{2}{{x}^{2}}[\/latex]<\/li>\n<li>[latex]f(x)=\\frac{ \\sin x}{x}[\/latex]<\/li>\n<li>[latex]f(x)={ \\tan }^{-1}(x)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043183885\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043183885\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165043183885\" style=\"list-style-type: lower-alpha\">\n<li>Using the algebraic limit laws, we have [latex]\\underset{x\\to \\infty }{\\text{lim}}(5-\\frac{2}{{x}^{2}})=\\underset{x\\to \\infty }{\\text{lim}}5-2(\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{x}).(\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{x})=5-2\u00b70=5.[\/latex]Similarly, [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=5.[\/latex]Therefore, [latex]f(x)=5-\\frac{2}{{x}^{2}}[\/latex] has a horizontal asymptote of [latex]y=5[\/latex] and [latex]f[\/latex] approaches this horizontal asymptote as [latex]x\\to \\text{\u00b1}\\infty[\/latex] as shown in the following graph.\n<div id=\"CNX_Calc_Figure_04_06_004\" class=\"wp-caption aligncenter\">\n<div style=\"width: 502px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211033\/CNX_Calc_Figure_04_06_003.jpg\" alt=\"The function f(x) = 5 \u2013 2\/x2 is graphed. The function approaches the horizontal asymptote y = 5 as x approaches \u00b1\u221e.\" width=\"492\" height=\"309\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4.<\/strong> This function approaches a horizontal asymptote as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>Since [latex]-1\\le \\sin x\\le 1[\/latex] for all [latex]x,[\/latex] we have\n<div id=\"fs-id1165043093355\" class=\"equation unnumbered\">[latex]\\frac{-1}{x}\\le \\frac{ \\sin x}{x}\\le \\frac{1}{x}[\/latex]<\/div>\n<p>for all [latex]x\\ne 0.[\/latex] Also, since<\/p>\n<div id=\"fs-id1165043197153\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{-1}{x}=0=\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{x},[\/latex]<\/div>\n<p>we can apply the squeeze theorem to conclude that<\/p>\n<div id=\"fs-id1165043036581\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{ \\sin x}{x}=0.[\/latex]<\/div>\n<p>Similarly,<\/p>\n<div id=\"fs-id1165043122536\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{ \\sin x}{x}=0.[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Thus, [latex]f(x)=\\frac{ \\sin x}{x}[\/latex] has a horizontal asymptote of [latex]y=0[\/latex] and [latex]f(x)[\/latex] approaches this horizontal asymptote as [latex]x\\to \\text{\u00b1}\\infty[\/latex] as shown in the following graph.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_005\" class=\"wp-caption aligncenter\">\n<div style=\"width: 727px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211036\/CNX_Calc_Figure_04_06_004.jpg\" alt=\"The function f(x) = (sin x)\/x is shown. It has a global maximum at (0, 1) and then proceeds to oscillate around y = 0 with decreasing amplitude.\" width=\"717\" height=\"193\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/strong> This function crosses its horizontal asymptote multiple times.<\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>To evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}{ \\tan }^{-1}(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{ \\tan }^{-1}(x),[\/latex] we first consider the graph of [latex]y= \\tan (x)[\/latex] over the interval [latex](\\text{\u2212}\\pi \\text{\/}2,\\pi \\text{\/}2)[\/latex] as shown in the following graph.\n<div id=\"CNX_Calc_Figure_04_06_006\" class=\"wp-caption aligncenter\"><span id=\"fs-id1165042710828\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211039\/CNX_Calc_Figure_04_06_021.jpg\" alt=\"The function f(x) = tan x is shown. It increases from (\u2212\u03c0\/2, \u2212\u221e), passes through the origin, and then increases toward (\u03c0\/2, \u221e). There are vertical dashed lines marking x = \u00b1\u03c0\/2.\" \/><\/span><\/div>\n<div class=\"wp-caption-text\">The graph of [latex]\\tan x[\/latex] has vertical asymptotes at [latex]x=\\text{\u00b1}\\frac{\\pi }{2}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165043092430\">Since<\/p>\n<div id=\"fs-id1165043119614\" class=\"equation unnumbered\">[latex]\\underset{x\\to {(\\pi \\text{\/}2)}^{-}}{\\text{lim}} \\tan x=\\infty ,[\/latex]<\/div>\n<p id=\"fs-id1165042514177\">it follows that<\/p>\n<div id=\"fs-id1165042563973\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}{ \\tan }^{-1}(x)=\\frac{\\pi }{2}.[\/latex]<\/div>\n<p id=\"fs-id1165043097156\">Similarly, since<\/p>\n<div id=\"fs-id1165042923290\" class=\"equation unnumbered\">[latex]\\underset{x\\to {(\\pi \\text{\/}2)}^{+}}{\\text{lim}} \\tan x=\\text{\u2212}\\infty ,[\/latex]<\/div>\n<p id=\"fs-id1165043131939\">it follows that<\/p>\n<div id=\"fs-id1165043056813\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{ \\tan }^{-1}(x)=-\\frac{\\pi }{2}.[\/latex]<\/div>\n<p id=\"fs-id1165042707528\">As a result, [latex]y=\\frac{\\pi }{2}[\/latex] and [latex]y=-\\frac{\\pi }{2}[\/latex] are horizontal asymptotes of [latex]f(x)={ \\tan }^{-1}(x)[\/latex] as shown in the following graph.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_007\" class=\"wp-caption aligncenter\">\n<div style=\"width: 501px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211043\/CNX_Calc_Figure_04_06_005.jpg\" alt=\"The function f(x) = tan\u22121 x is shown. It increases from (\u2212\u221e, \u2212\u03c0\/2), passes through the origin, and then increases toward (\u221e, \u03c0\/2). There are horizontal dashed lines marking y = \u00b1\u03c0\/2.\" width=\"491\" height=\"199\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7.<\/strong> This function has two horizontal asymptotes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042320881\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042320884\" class=\"exercise\">\n<div id=\"fs-id1165043315933\" class=\"textbox\">\n<p id=\"fs-id1165043315935\">Evaluate [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}(3+\\frac{4}{x})[\/latex] and [latex]\\underset{x\\to \\infty }{\\text{lim}}(3+\\frac{4}{x}).[\/latex] Determine the horizontal asymptotes of [latex]f(x)=3+\\frac{4}{x},[\/latex] if any.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043390798\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043390798\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043390798\">Both limits are 3. The line [latex]y=3[\/latex] is a horizontal asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165042318505\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042318511\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}1\\text{\/}x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042333169\" class=\"bc-section section\">\n<h2>Infinite Limits at Infinity<\/h2>\n<p id=\"fs-id1165042333174\">Sometimes the values of a function [latex]f[\/latex] become arbitrarily large as [latex]x\\to \\infty[\/latex] (or as [latex]x\\to \\text{\u2212}\\infty ).[\/latex] In this case, we write [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty[\/latex] (or [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=\\infty ).[\/latex] On the other hand, if the values of [latex]f[\/latex] are negative but become arbitrarily large in magnitude as [latex]x\\to \\infty[\/latex] (or as [latex]x\\to \\text{\u2212}\\infty ),[\/latex] we write [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty[\/latex] (or [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty ).[\/latex]<\/p>\n<p id=\"fs-id1165042606820\">For example, consider the function [latex]f(x)={x}^{3}.[\/latex] As seen in <a class=\"autogenerated-content\" href=\"#fs-id1165042406634\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_008\">(Figure)<\/a>, as [latex]x\\to \\infty[\/latex] the values [latex]f(x)[\/latex] become arbitrarily large. Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{3}=\\infty .[\/latex] On the other hand, as [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the values of [latex]f(x)={x}^{3}[\/latex] are negative but become arbitrarily large in magnitude. Consequently, [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{3}=\\text{\u2212}\\infty .[\/latex]<\/p>\n<table id=\"fs-id1165042406634\" class=\"column-header\" summary=\"The table has four rows and six columns. The first column is a header column and it reads x, x3, x, and x3. After the header, the first row reads 10, 20, 50, 100, and 1000. The second row reads 1000, 8000, 125000, 1,000,000, and 1,000,000,000. The third row reads \u221210, \u221220, \u221250, \u2212100, and \u22121000. The forth row reads \u22121000, \u22128000, \u2212125,000, \u22121,000,000, and \u22121,000,000,000.\">\n<caption>Values of a power function as [latex]x\\to \\text{\u00b1}\\infty[\/latex]<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>10<\/td>\n<td>20<\/td>\n<td>50<\/td>\n<td>100<\/td>\n<td>1000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]{x}^{3}[\/latex]<\/strong><\/td>\n<td>1000<\/td>\n<td>8000<\/td>\n<td>125,000<\/td>\n<td>1,000,000<\/td>\n<td>1,000,000,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>-10<\/td>\n<td>-20<\/td>\n<td>-50<\/td>\n<td>-100<\/td>\n<td>-1000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]{x}^{3}[\/latex]<\/strong><\/td>\n<td>-1000<\/td>\n<td>-8000<\/td>\n<td>-125,000<\/td>\n<td>-1,000,000<\/td>\n<td>-1,000,000,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"CNX_Calc_Figure_04_06_008\" class=\"wp-caption aligncenter\">\n<div style=\"width: 652px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211045\/CNX_Calc_Figure_04_06_022.jpg\" alt=\"The function f(x) = x3 is graphed. It is apparent that this function rapidly approaches infinity as x approaches infinity.\" width=\"642\" height=\"272\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 8.<\/strong> For this function, the functional values approach infinity as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043276353\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1165043276356\">(Informal) We say a function [latex]f[\/latex] has an infinite limit at infinity and write<\/p>\n<div id=\"fs-id1165043276364\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty .[\/latex]<\/div>\n<p id=\"fs-id1165042709557\">if [latex]f(x)[\/latex] becomes arbitrarily large for [latex]x[\/latex] sufficiently large. We say a function has a negative infinite limit at infinity and write<\/p>\n<div id=\"fs-id1165042647077\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty .[\/latex]<\/div>\n<p id=\"fs-id1165042327355\">if [latex]f(x)<0[\/latex] and [latex]|f(x)|[\/latex] becomes arbitrarily large for [latex]x[\/latex] sufficiently large. Similarly, we can define infinite limits as [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042328702\" class=\"bc-section section\">\n<h2>Formal Definitions<\/h2>\n<p id=\"fs-id1165042328707\">Earlier, we used the terms <em>arbitrarily close<\/em>, <em>arbitrarily large<\/em>, and <em>sufficiently large<\/em> to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.<\/p>\n<div id=\"fs-id1165043308442\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1165043308445\">(Formal) We say a function [latex]f[\/latex] has a limit at infinity, if there exists a real number [latex]L[\/latex] such that for all [latex]\\epsilon >0,[\/latex] there exists [latex]N>0[\/latex] such that<\/p>\n<div id=\"fs-id1165043395062\" class=\"equation unnumbered\">[latex]|f(x)-L|<\\epsilon[\/latex]<\/div>\n<p id=\"fs-id1165043298558\">for all [latex]x>N.[\/latex] In that case, we write<\/p>\n<div id=\"fs-id1165042364605\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L[\/latex]<\/div>\n<p id=\"fs-id1165042512686\">(see <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_009\">(Figure)<\/a>).<\/p>\n<p id=\"fs-id1165042327662\">We say a function [latex]f[\/latex] has a limit at negative infinity if there exists a real number [latex]L[\/latex] such that for all [latex]\\epsilon >0,[\/latex] there exists [latex]N<0[\/latex] such that<\/p>\n<div id=\"fs-id1165042331766\" class=\"equation unnumbered\">[latex]|f(x)-L|<\\epsilon[\/latex]<\/div>\n<p id=\"fs-id1165042472034\">for all [latex]x<N.[\/latex] In that case, we write<\/p>\n<div id=\"fs-id1165042472050\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=L.[\/latex]<\/div>\n<\/div>\n<div id=\"CNX_Calc_Figure_04_06_009\" class=\"wp-caption aligncenter\">\n<div style=\"width: 379px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211049\/CNX_Calc_Figure_04_06_023.jpg\" alt=\"The function f(x) is graphed, and it has a horizontal asymptote at L. L is marked on the y axis, as is L + \u0949 and L \u2013 \u0949. On the x axis, N is marked as the value of x such that f(x) = L + \u0949.\" width=\"369\" height=\"278\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 9.<\/strong> For a function with a limit at infinity, for all [latex]x&gt;N,[\/latex] [latex]|f(x)-L|&lt;\\epsilon .[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043396243\">Earlier in this section, we used graphical evidence in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_001\">(Figure)<\/a> and numerical evidence in <a class=\"autogenerated-content\" href=\"#fs-id1165043428402\">(Figure)<\/a> to conclude that [latex]\\underset{x\\to \\infty }{\\text{lim}}(\\frac{2+1}{x})=2.[\/latex] Here we use the formal definition of limit at infinity to prove this result rigorously.<\/p>\n<div class=\"textbox examples\">\n<h3>A Finite Limit at Infinity Example<\/h3>\n<div id=\"fs-id1165042587292\" class=\"exercise\">\n<div id=\"fs-id1165042587294\" class=\"textbox\">\n<p id=\"fs-id1165042587296\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\text{lim}}(\\frac{2+1}{x})=2.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042369578\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042369578\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042369578\">Let [latex]\\epsilon >0.[\/latex] Let [latex]N=\\frac{1}{\\epsilon }.[\/latex] Therefore, for all [latex]x>N,[\/latex] we have<\/p>\n<div id=\"fs-id1165043312498\" class=\"equation unnumbered\">[latex]|2+\\frac{1}{x}-2|=|\\frac{1}{x}|=\\frac{1}{x}<\\frac{1}{N}=\\epsilon \\text{.}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042480092\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042480095\" class=\"exercise\">\n<div id=\"fs-id1165042480097\" class=\"textbox\">\n<p id=\"fs-id1165042480099\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\text{lim}}(\\frac{3-1}{{x}^{2}})=3.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042367887\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042367887\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042367887\">Let [latex]\\epsilon >0.[\/latex] Let [latex]N=\\frac{1}{\\sqrt{\\epsilon }}.[\/latex] Therefore, for all [latex]x>N,[\/latex] we have<\/p>\n<p id=\"fs-id1165042376362\">[latex]|3-\\frac{1}{{x}^{2}}-3|=\\frac{1}{{x}^{2}}<\\frac{1}{{N}^{2}}=\\epsilon[\/latex]<\/p>\n<p id=\"fs-id1165042320298\">Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}(3-1\\text{\/}{x}^{2})=3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042332059\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042332065\">Let [latex]N=\\frac{1}{\\sqrt{\\epsilon }}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042374773\">We now turn our attention to a more precise definition for an infinite limit at infinity.<\/p>\n<div id=\"fs-id1165042374776\" class=\"textbox key-takeaways\">\n<div class=\"title\">\n<h3>Definition<\/h3>\n<\/div>\n<p id=\"fs-id1165042374780\">(Formal) We say a function [latex]f[\/latex] has an infinite limit at infinity and write<\/p>\n<div id=\"fs-id1165042364247\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty[\/latex]<\/div>\n<p id=\"fs-id1165043423999\">if for all [latex]M>0,[\/latex] there exists an [latex]N>0[\/latex] such that<\/p>\n<div id=\"fs-id1165043248795\" class=\"equation unnumbered\">[latex]f(x)>M[\/latex]<\/div>\n<p id=\"fs-id1165042374733\">for all [latex]x>N[\/latex] (see <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_010\">(Figure)<\/a>).<\/p>\n<p id=\"fs-id1165042374750\">We say a function has a negative infinite limit at infinity and write<\/p>\n<div id=\"fs-id1165042374753\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty[\/latex]<\/div>\n<p id=\"fs-id1165043426267\">if for all [latex]M<0,[\/latex] there exists an [latex]N>0[\/latex] such that<\/p>\n<div id=\"fs-id1165043259687\" class=\"equation unnumbered\">[latex]f(x)<M[\/latex]<\/div>\n<p id=\"fs-id1165043259707\">for all [latex]x>N.[\/latex]<\/p>\n<p id=\"fs-id1165043259751\">Similarly we can define limits as [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\n<\/div>\n<div id=\"CNX_Calc_Figure_04_06_010\" class=\"wp-caption aligncenter\">\n<div style=\"width: 466px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211052\/CNX_Calc_Figure_04_06_024.jpg\" alt=\"The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M.\" width=\"456\" height=\"315\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 10.<\/strong> For a function with an infinite limit at infinity, for all [latex]x&gt;N,[\/latex] [latex]f(x)&gt;M.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042705963\">Earlier, we used graphical evidence (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_008\">(Figure)<\/a>) and numerical evidence (<a class=\"autogenerated-content\" href=\"#fs-id1165042406634\">(Figure)<\/a>) to conclude that [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{3}=\\infty .[\/latex] Here we use the formal definition of infinite limit at infinity to prove that result.<\/p>\n<div id=\"fs-id1165042323534\" class=\"textbox examples\">\n<h3>An Infinite Limit at Infinity<\/h3>\n<div id=\"fs-id1165042323536\" class=\"exercise\">\n<div id=\"fs-id1165042323538\" class=\"textbox\">\n<p id=\"fs-id1165043395589\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{3}=\\infty .[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043430975\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043430975\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043430975\">Let [latex]M>0.[\/latex] Let [latex]N=\\sqrt[3]{M}.[\/latex] Then, for all [latex]x>N,[\/latex] we have<\/p>\n<div id=\"fs-id1165043174087\" class=\"equation unnumbered\">[latex]{x}^{3}>{N}^{3}={(\\sqrt[3]{M})}^{3}=M.[\/latex]<\/div>\n<p id=\"fs-id1165042604681\">Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{3}=\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042323710\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042320222\" class=\"exercise\">\n<div id=\"fs-id1165042320224\" class=\"textbox\">\n<p id=\"fs-id1165042320226\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\text{lim}}3{x}^{2}=\\infty .[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042708272\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042708272\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042708272\">Let [latex]M>0.[\/latex] Let [latex]N=\\sqrt{\\frac{M}{3}}.[\/latex] Then, for all [latex]x>N,[\/latex] we have<\/p>\n<p id=\"fs-id1165042383154\">[latex]3{x}^{2}>3{N}^{2}=3{(\\sqrt{\\frac{M}{3}})}^{2}{2}^{}=\\frac{3M}{3}=M[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043219098\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043219104\">Let [latex]N=\\sqrt{\\frac{M}{3}}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042368487\" class=\"bc-section section\">\n<h1>End Behavior<\/h1>\n<p id=\"fs-id1165042368492\">The behavior of a function as [latex]x\\to \\text{\u00b1}\\infty[\/latex] is called the function\u2019s <strong>end behavior<\/strong>. At each of the function\u2019s ends, the function could exhibit one of the following types of behavior:<\/p>\n<ol id=\"fs-id1165042349939\">\n<li>The function [latex]f(x)[\/latex] approaches a horizontal asymptote [latex]y=L.[\/latex]<\/li>\n<li>The function [latex]f(x)\\to \\infty[\/latex] or [latex]f(x)\\to \\text{\u2212}\\infty .[\/latex]<\/li>\n<li>The function does not approach a finite limit, nor does it approach [latex]\\infty[\/latex] or [latex]\\text{\u2212}\\infty .[\/latex] In this case, the function may have some oscillatory behavior.<\/li>\n<\/ol>\n<p id=\"fs-id1165042323661\">Let\u2019s consider several classes of functions here and look at the different types of end behaviors for these functions.<\/p>\n<div id=\"fs-id1165042323666\" class=\"bc-section section\">\n<h2>End Behavior for Polynomial Functions<\/h2>\n<p id=\"fs-id1165042323672\">Consider the power function [latex]f(x)={x}^{n}[\/latex] where [latex]n[\/latex] is a positive integer. From <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_011\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_012\">(Figure)<\/a>, we see that<\/p>\n<div id=\"fs-id1165042545843\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}=\\infty ;n=1,2,3\\text{,\u2026}[\/latex]<\/div>\n<p>and<\/p>\n<div id=\"fs-id1165042705928\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}=\\bigg\\{\\begin{array}{c}\\infty ;n=2,4,6\\text{,\u2026}\\hfill \\\\ \\text{\u2212}\\infty ;n=1,3,5\\text{,\u2026}\\hfill \\end{array}.[\/latex]<\/div>\n<div id=\"CNX_Calc_Figure_04_06_011\" class=\"wp-caption aligncenter\">\n<div style=\"width: 435px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211055\/CNX_Calc_Figure_04_06_025.jpg\" alt=\"The functions x2, x4, and x6 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.\" width=\"425\" height=\"358\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 11.<\/strong> For power functions with an even power of [latex]n,[\/latex] [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}=\\infty =\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"CNX_Calc_Figure_04_06_012\" class=\"wp-caption aligncenter\">\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211058\/CNX_Calc_Figure_04_06_026.jpg\" alt=\"The functions x, x3, and x5 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.\" width=\"417\" height=\"352\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 12.<\/strong> For power functions with an odd power of [latex]n,[\/latex] [latex]\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}=\\infty [\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}=\\text{\u2212}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042318644\">Using these facts, it is not difficult to evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}c{x}^{n}[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}c{x}^{n},[\/latex] where [latex]c[\/latex] is any constant and [latex]n[\/latex] is a positive integer. If [latex]c>0,[\/latex] the graph of [latex]y=c{x}^{n}[\/latex] is a vertical stretch or compression of [latex]y={x}^{n},[\/latex] and therefore<\/p>\n<div id=\"fs-id1165042327426\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}c{x}^{n}=\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}\\text{ and }\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}c{x}^{n}=\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}\\text{ if }c>0.[\/latex]<\/div>\n<p id=\"fs-id1165043424818\">If [latex]c<0,[\/latex] the graph of [latex]y=c{x}^{n}[\/latex] is a vertical stretch or compression combined with a reflection about the [latex]x[\/latex]-axis, and therefore<\/p>\n<div id=\"fs-id1165042327325\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}c{x}^{n}=\\text{\u2212}\\underset{x\\to \\infty }{\\text{lim}}{x}^{n}\\text{ and }\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}c{x}^{n}=\\text{\u2212}\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{n}\\text{ if }c<0.[\/latex]<\/div>\n<p id=\"fs-id1165042640745\">If [latex]c=0,y=c{x}^{n}=0,[\/latex] in which case [latex]\\underset{x\\to \\infty }{\\text{lim}}c{x}^{n}=0=\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}c{x}^{n}.[\/latex]<\/p>\n<div id=\"fs-id1165043219126\" class=\"textbox examples\">\n<h3>Limits at Infinity for Power Functions<\/h3>\n<div id=\"fs-id1165043219128\" class=\"exercise\">\n<div id=\"fs-id1165043219130\" class=\"textbox\">\n<p>For each function [latex]f,[\/latex] evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x).[\/latex]<\/p>\n<ol id=\"fs-id1165042333246\" style=\"list-style-type: lower-alpha\">\n<li>[latex]f(x)=-5{x}^{3}[\/latex]<\/li>\n<li>[latex]f(x)=2{x}^{4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043254252\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043254252\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165043254252\" style=\"list-style-type: lower-alpha\">\n<li>Since the coefficient of [latex]{x}^{3}[\/latex] is -5, the graph of [latex]f(x)=-5{x}^{3}[\/latex] involves a vertical stretch and reflection of the graph of [latex]y={x}^{3}[\/latex] about the [latex]x[\/latex]-axis. Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}(-5{x}^{3})=\\text{\u2212}\\infty[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}(-5{x}^{3})=\\infty .[\/latex]<\/li>\n<li>Since the coefficient of [latex]{x}^{4}[\/latex] is 2, the graph of [latex]f(x)=2{x}^{4}[\/latex] is a vertical stretch of the graph of [latex]y={x}^{4}.[\/latex] Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}2{x}^{4}=\\infty[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}2{x}^{4}=\\infty .[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042401057\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042401060\" class=\"exercise\">\n<div id=\"fs-id1165042401062\" class=\"textbox\">\n<p id=\"fs-id1165042401064\">Let [latex]f(x)=-3{x}^{4}.[\/latex] Find [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042708212\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042708212\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042708212\">[latex]\\text{\u2212}\\infty[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042708221\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042708228\">The coefficient -3 is negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042708240\">We now look at how the limits at infinity for power functions can be used to determine [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}f(x)[\/latex] for any polynomial function [latex]f.[\/latex] Consider a polynomial function<\/p>\n<div id=\"fs-id1165042710943\" class=\"equation unnumbered\">[latex]f(x)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\\text{\u2026}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"fs-id1165043327638\">of degree [latex]n\\ge 1[\/latex] so that [latex]{a}_{n}\\ne 0.[\/latex] Factoring, we see that<\/p>\n<div id=\"fs-id1165042319257\" class=\"equation unnumbered\">[latex]f(x)={a}_{n}{x}^{n}(1+\\frac{{a}_{n-1}}{{a}_{n}}\\frac{1}{x}+\\text{\u2026}+\\frac{{a}_{1}}{{a}_{n}}\\frac{1}{{x}^{n-1}}+\\frac{{a}_{0}}{{a}_{n}}).[\/latex]<\/div>\n<p id=\"fs-id1165043348532\">As [latex]x\\to \\text{\u00b1}\\infty ,[\/latex] all the terms inside the parentheses approach zero except the first term. We conclude that<\/p>\n<div id=\"fs-id1165043348550\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}f(x)=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}{a}_{n}{x}^{n}.[\/latex]<\/div>\n<p id=\"fs-id1165043317360\">For example, the function [latex]f(x)=5{x}^{3}-3{x}^{2}+4[\/latex] behaves like [latex]g(x)=5{x}^{3}[\/latex] as [latex]x\\to \\text{\u00b1}\\infty[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_013\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165043250976\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_013\" class=\"wp-caption aligncenter\">\n<div style=\"width: 392px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211101\/CNX_Calc_Figure_04_06_006.jpg\" alt=\"Both functions f(x) = 5x3 \u2013 3x2 + 4 and g(x) = 5x3 are plotted. Their behavior for large positive and large negative numbers converges.\" width=\"382\" height=\"272\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 13.<\/strong> The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.<\/p>\n<\/div>\n<\/div>\n<table id=\"fs-id1165043250976\" class=\"column-header\" summary=\"The table has six rows and four columns. The first column is a header column and it reads x, f(x) = 5x3 \u2013 3x2 + 4, g(x) = 5x3, x, f(x) = 5x3 \u2013 3x2 + 4, and g(x) = 5x3. After the header, the first row reads 10, 100, and 1000. The second row reads 4704, 4,970,004, and 4,997,000,004. The third row reads 5000, 5,000,000, 5,000,000,000. The fourth row reads \u221210, \u2212100, and \u22121000. The fifth row reads \u22125296, \u22125,029,996, and \u22125,002,999,996. The sixth row reads \u22125000, \u22125,000,000, and \u22125,000,000,000.\">\n<caption>A polynomial\u2019s end behavior is determined by the term with the largest exponent.<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>10<\/td>\n<td>100<\/td>\n<td>1000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]f(x)=5{x}^{3}-3{x}^{2}+4[\/latex]<\/strong><\/td>\n<td>4704<\/td>\n<td>4,970,004<\/td>\n<td>4,997,000,004<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]g(x)=5{x}^{3}[\/latex]<\/strong><\/td>\n<td>5000<\/td>\n<td>5,000,000<\/td>\n<td>5,000,000,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>-10<\/td>\n<td>-100<\/td>\n<td>-1000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]f(x)=5{x}^{3}-3{x}^{2}+4[\/latex]<\/strong><\/td>\n<td>-5296<\/td>\n<td>-5,029,996<\/td>\n<td>-5,002,999,996<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]g(x)=5{x}^{3}[\/latex]<\/strong><\/td>\n<td>-5000<\/td>\n<td>-5,000,000<\/td>\n<td>-5,000,000,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"bc-section section\">\n<h2>End Behavior for Algebraic Functions<\/h2>\n<p id=\"fs-id1165042638493\">The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In <a class=\"autogenerated-content\" href=\"#fs-id1165042638553\">(Figure)<\/a>, we show that the limits at infinity of a rational function [latex]f(x)=\\frac{p(x)}{q(x)}[\/latex] depend on the relationship between the degree of the numerator and the degree of the denominator. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of [latex]x[\/latex] appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of [latex]x.[\/latex]<\/p>\n<div id=\"fs-id1165042638553\" class=\"textbox examples\">\n<h3>Determining End Behavior for Rational Functions<\/h3>\n<div id=\"fs-id1165042638555\" class=\"exercise\">\n<div id=\"fs-id1165042638557\" class=\"textbox\">\n<p id=\"fs-id1165042638562\">For each of the following functions, determine the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \\text{\u2212}\\infty .[\/latex] Then, use this information to describe the end behavior of the function.<\/p>\n<ol id=\"fs-id1165043390828\" style=\"list-style-type: lower-alpha\">\n<li>[latex]f(x)=\\frac{3x-1}{2x+5}[\/latex] (<em>Note:<\/em> The degree of the numerator and the denominator are the same.)<\/li>\n<li>[latex]f(x)=\\frac{3{x}^{2}+2x}{4{x}^{3}-5x+7}[\/latex] (<em>Note:<\/em> The degree of numerator is less than the degree of the denominator.)<\/li>\n<li>[latex]f(x)=\\frac{3{x}^{2}+4x}{x+2}[\/latex] (<em>Note:<\/em> The degree of numerator is greater than the degree of the denominator.)<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042708379\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042708379\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165042708379\" style=\"list-style-type: lower-alpha\">\n<li>The highest power of [latex]x[\/latex] in the denominator is [latex]x.[\/latex] Therefore, dividing the numerator and denominator by [latex]x[\/latex] and applying the algebraic limit laws, we see that\n<div id=\"fs-id1165043281584\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3x-1}{2x+5}& =\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3-1\\text{\/}x}{2+5\\text{\/}x}\\hfill \\\\ & =\\frac{\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}(3-1\\text{\/}x)}{\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}(2+5\\text{\/}x)}\\hfill \\\\ & =\\frac{\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}3-\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}1\\text{\/}x}{\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}2+\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}5\\text{\/}x}\\hfill \\\\ & =\\frac{3-0}{2+0}=\\frac{3}{2}.\\hfill \\end{array}[\/latex]<\/div>\n<p>Since [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}f(x)=\\frac{3}{2},[\/latex] we know that [latex]y=\\frac{3}{2}[\/latex] is a horizontal asymptote for this function as shown in the following graph.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_014\" class=\"wp-caption aligncenter\">\n<div style=\"width: 352px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211104\/CNX_Calc_Figure_04_06_007.jpg\" alt=\"The function f(x) = (3x + 1)\/(2x + 5) is plotted as is its horizontal asymptote at y = 3\/2.\" width=\"342\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 14.<\/strong> The graph of this rational function approaches a horizontal asymptote as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>Since the largest power of [latex]x[\/latex] appearing in the denominator is [latex]{x}^{3},[\/latex] divide the numerator and denominator by [latex]{x}^{3}.[\/latex] After doing so and applying algebraic limit laws, we obtain\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3{x}^{2}+2x}{4{x}^{3}-5x+7}=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3\\text{\/}x+2\\text{\/}{x}^{2}}{4-5\\text{\/}{x}^{2}+7\\text{\/}{x}^{3}}=\\frac{3.0+2.0}{4-5.0+7.0}=0.[\/latex]<\/div>\n<p>Therefore [latex]f[\/latex] has a horizontal asymptote of [latex]y=0[\/latex] as shown in the following graph.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_015\" class=\"wp-caption aligncenter\">\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211107\/CNX_Calc_Figure_04_06_008.jpg\" alt=\"The function f(x) = (3x2 + 2x)\/(4x2 \u2013 5x + 7) is plotted as is its horizontal asymptote at y = 0.\" width=\"417\" height=\"422\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 15.<\/strong> The graph of this rational function approaches the horizontal asymptote [latex]y=0[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>Dividing the numerator and denominator by [latex]x,[\/latex] we have\n<div id=\"fs-id1165042333346\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3{x}^{2}+4x}{x+2}=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3x+4}{1+2\\text{\/}x}.[\/latex]<\/div>\n<p>As [latex]x\\to \\text{\u00b1}\\infty ,[\/latex] the denominator approaches 1. As [latex]x\\to \\infty ,[\/latex] the numerator approaches [latex]+\\infty .[\/latex] As [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the numerator approaches [latex]\\text{\u2212}\\infty .[\/latex] Therefore [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty ,[\/latex] whereas [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty[\/latex] as shown in the following figure.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_016\" class=\"wp-caption aligncenter\">\n<div style=\"width: 579px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211110\/CNX_Calc_Figure_04_06_027.jpg\" alt=\"The function f(x) = (3x2 + 4x)\/(x + 2) is plotted. It appears to have a diagonal asymptote as well as a vertical asymptote at x = \u22122.\" width=\"569\" height=\"497\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 16.<\/strong> As [latex]x\\to \\infty ,[\/latex] the values [latex]f(x)\\to \\infty .[\/latex] As [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the values [latex]f(x)\\to \\text{\u2212}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042660288\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042660292\" class=\"exercise\">\n<div id=\"fs-id1165042660294\" class=\"textbox\">\n<p id=\"fs-id1165042660296\">Evaluate [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{3{x}^{2}+2x-1}{5{x}^{2}-4x+7}[\/latex] and use these limits to determine the end behavior of [latex]f(x)=\\frac{3{x}^{2}+2x-2}{5{x}^{2}-4x+7}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042374871\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042374871\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042374871\">[latex]\\frac{3}{5}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042374882\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042374889\">Divide the numerator and denominator by [latex]{x}^{2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042374905\">Before proceeding, consider the graph of [latex]f(x)=\\frac{(3{x}^{2}+4x)}{(x+2)}[\/latex] shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_017\">(Figure)<\/a>. As [latex]x\\to \\infty[\/latex] and [latex]x\\to \\text{\u2212}\\infty ,[\/latex] the graph of [latex]f[\/latex] appears almost linear. Although [latex]f[\/latex] is certainly not a linear function, we now investigate why the graph of [latex]f[\/latex] seems to be approaching a linear function. First, using long division of polynomials, we can write<\/p>\n<div id=\"fs-id1165043219202\" class=\"equation unnumbered\">[latex]f(x)=\\frac{3{x}^{2}+4x}{x+2}=3x-2+\\frac{4}{x+2}.[\/latex]<\/div>\n<p id=\"fs-id1165043219268\">Since [latex]\\frac{4}{(x+2)}\\to 0[\/latex] as [latex]x\\to \\text{\u00b1}\\infty ,[\/latex] we conclude that<\/p>\n<div id=\"fs-id1165042465555\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}(f(x)-(3x-2))=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{4}{x+2}=0.[\/latex]<\/div>\n<p id=\"fs-id1165042465646\">Therefore, the graph of [latex]f[\/latex] approaches the line [latex]y=3x-2[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] This line is known as an <strong>oblique asymptote<\/strong> for [latex]f[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_017\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_04_06_017\" class=\"wp-caption aligncenter\">\n<div style=\"width: 277px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211113\/CNX_Calc_Figure_04_06_009.jpg\" alt=\"The function f(x) = (3x2 + 4x)\/(x + 2) is plotted as is its diagonal asymptote y = 3x \u2013 2.\" width=\"267\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 17. The graph of the rational function [latex]f(x)=(3{x}^{2}+4x)\\text{\/}(x+2)[\/latex] approaches the oblique asymptote [latex]y=3x-2\\text{as}x\\to \\text{\u00b1}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042461217\">We can summarize the results of <a class=\"autogenerated-content\" href=\"#fs-id1165042638553\">(Figure)<\/a> to make the following conclusion regarding end behavior for rational functions. Consider a rational function<\/p>\n<div id=\"fs-id1165042461226\" class=\"equation unnumbered\">[latex]f(x)=\\frac{p(x)}{q(x)}=\\frac{{a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\\text{\u2026}+{a}_{1}x+{a}_{0}}{{b}_{m}{x}^{m}+{b}_{m-1}{x}^{m-1}+\\text{\u2026}+{b}_{1}x+{b}_{0}},[\/latex]<\/div>\n<p id=\"fs-id1165042315666\">where [latex]{a}_{n}\\ne 0\\text{ and }{b}_{m}\\ne 0.[\/latex]<\/p>\n<ol id=\"fs-id1165043422338\">\n<li>If the degree of the numerator is the same as the degree of the denominator [latex](n=m),[\/latex] then [latex]f[\/latex] has a horizontal asymptote of [latex]y={a}_{n}\\text{\/}{b}_{m}[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/li>\n<li>If the degree of the numerator is less than the degree of the denominator [latex](n<m),[\/latex] then [latex]f[\/latex] has a horizontal asymptote of [latex]y=0[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/li>\n<li>If the degree of the numerator is greater than the degree of the denominator [latex](n>m),[\/latex] then [latex]f[\/latex] does not have a horizontal asymptote. The limits at infinity are either positive or negative infinity, depending on the signs of the leading terms. In addition, using long division, the function can be rewritten as\n<div id=\"fs-id1165043422484\" class=\"equation unnumbered\">[latex]f(x)=\\frac{p(x)}{q(x)}=g(x)+\\frac{r(x)}{q(x)},[\/latex]<\/div>\n<p>where the degree of [latex]r(x)[\/latex] is less than the degree of [latex]q(x).[\/latex] As a result, [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}r(x)\\text{\/}q(x)=0.[\/latex] Therefore, the values of [latex]\\left[f(x)-g(x)\\right][\/latex] approach zero as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] If the degree of [latex]p(x)[\/latex] is exactly one more than the degree of [latex]q(x)[\/latex] [latex](n=m+1),[\/latex] the function [latex]g(x)[\/latex] is a linear function. In this case, we call [latex]g(x)[\/latex] an oblique asymptote.<br \/>\nNow let\u2019s consider the end behavior for functions involving a radical.<\/li>\n<\/ol>\n<div id=\"fs-id1165042631816\" class=\"textbox examples\">\n<h3>Determining End Behavior for a Function Involving a Radical<\/h3>\n<div id=\"fs-id1165042631818\" class=\"exercise\">\n<div id=\"fs-id1165042631820\" class=\"textbox\">\n<p id=\"fs-id1165042631826\">Find the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \\text{\u2212}\\infty[\/latex] for [latex]f(x)=\\frac{3x-2}{\\sqrt{4{x}^{2}+5}}[\/latex] and describe the end behavior of [latex]f.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042631905\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042631905\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042631905\">Let\u2019s use the same strategy as we did for rational functions: divide the numerator and denominator by a power of [latex]x.[\/latex] To determine the appropriate power of [latex]x,[\/latex] consider the expression [latex]\\sqrt{4{x}^{2}+5}[\/latex] in the denominator. Since<\/p>\n<div id=\"fs-id1165042418061\" class=\"equation unnumbered\">[latex]\\sqrt{4{x}^{2}+5}\\approx \\sqrt{4{x}^{2}}=2|x|[\/latex]<\/div>\n<p id=\"fs-id1165042418106\">for large values of [latex]x[\/latex] in effect [latex]x[\/latex] appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by [latex]|x|.[\/latex] Then, using the fact that [latex]|x|=x[\/latex] for [latex]x>0,[\/latex] [latex]|x|=\\text{\u2212}x[\/latex] for [latex]x<0,[\/latex] and [latex]|x|=\\sqrt{{x}^{2}}[\/latex] for all [latex]x,[\/latex] we calculate the limits as follows:<\/p>\n<div id=\"fs-id1165042418216\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\underset{x\\to \\infty }{\\text{lim}}\\frac{3x-2}{\\sqrt{4{x}^{2}+5}}& =\\hfill & \\underset{x\\to \\infty }{\\text{lim}}\\frac{(1\\text{\/}|x|)(3x-2)}{(1\\text{\/}|x|)\\sqrt{4{x}^{2}+5}}\\hfill \\\\ & =\\hfill & \\underset{x\\to \\infty }{\\text{lim}}\\frac{(1\\text{\/}x)(3x-2)}{\\sqrt{(1\\text{\/}{x}^{2})(4{x}^{2}+5)}}\\hfill \\\\ & =\\hfill & \\underset{x\\to \\infty }{\\text{lim}}\\frac{3-2\\text{\/}x}{\\sqrt{4+5\\text{\/}{x}^{2}}}=\\frac{3}{\\sqrt{4}}=\\frac{3}{2}\\hfill \\\\ \\hfill \\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{3x-2}{\\sqrt{4{x}^{2}+5}}& =\\hfill & \\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{(1\\text{\/}|x|)(3x-2)}{(1\\text{\/}|x|)\\sqrt{4{x}^{2}+5}}\\hfill \\\\ & =\\hfill & \\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{(-1\\text{\/}x)(3x-2)}{\\sqrt{(1\\text{\/}{x}^{2})(4{x}^{2}+5)}}\\hfill \\\\ & =\\hfill & \\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{-3+2\\text{\/}x}{\\sqrt{4+5\\text{\/}{x}^{2}}}=\\frac{-3}{\\sqrt{4}}=\\frac{-3}{2}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165042463819\">Therefore, [latex]f(x)[\/latex] approaches the horizontal asymptote [latex]y=\\frac{3}{2}[\/latex] as [latex]x\\to \\infty[\/latex] and the horizontal asymptote [latex]y=-\\frac{3}{2}[\/latex] as [latex]x\\to \\text{\u2212}\\infty[\/latex] as shown in the following graph.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_018\" class=\"wp-caption aligncenter\">\n<div style=\"width: 602px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211116\/CNX_Calc_Figure_04_06_010.jpg\" alt=\"The function f(x) = (3x \u2212 2)\/(the square root of the quantity (4x2 + 5)) is plotted. It has two horizontal asymptotes at y = \u00b13\/2, and it crosses y = \u22123\/2 before converging toward it from below.\" width=\"592\" height=\"197\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 18.<\/strong> This function has two horizontal asymptotes and it crosses one of the asymptotes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042463914\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042463919\" class=\"exercise\">\n<div id=\"fs-id1165042463921\" class=\"textbox\">\n<p id=\"fs-id1165042463923\">Evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{\\sqrt{3{x}^{2}+4}}{x+6}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043327293\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043327293\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043327293\">[latex]\\text{\u00b1}\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043327303\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043327311\">Divide the numerator and denominator by [latex]|x|.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043327330\" class=\"bc-section section\">\n<h2>Determining End Behavior for Transcendental Functions<\/h2>\n<p id=\"fs-id1165043327335\">The six basic trigonometric functions are periodic and do not approach a finite limit as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] For example, [latex]\\sin x[\/latex] oscillates between [latex]1\\text{ and }-1[\/latex] (<a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_019\">(Figure)<\/a>). The tangent function [latex]x[\/latex] has an infinite number of vertical asymptotes as [latex]x\\to \\text{\u00b1}\\infty ;[\/latex] therefore, it does not approach a finite limit nor does it approach [latex]\\text{\u00b1}\\infty[\/latex] as [latex]x\\to \\text{\u00b1}\\infty[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_020\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Calc_Figure_04_06_019\" class=\"wp-caption aligncenter\">\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211121\/CNX_Calc_Figure_04_06_011.jpg\" alt=\"The function f(x) = sin x is graphed.\" width=\"417\" height=\"197\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 19.<\/strong> The function [latex]f(x)= \\sin x[\/latex] oscillates between [latex]1\\text{ and }-1[\/latex] as [latex]x\\to \\text{\u00b1}\\infty [\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"CNX_Calc_Figure_04_06_020\" class=\"wp-caption aligncenter\">\n<div style=\"width: 510px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211123\/CNX_Calc_Figure_04_06_012.jpg\" alt=\"The function f(x) = tan x is graphed.\" width=\"500\" height=\"272\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 20.<\/strong> The function [latex]f(x)= \\tan x[\/latex] does not approach a limit and does not approach [latex]\\text{\u00b1}\\infty [\/latex] as [latex]x\\to \\text{\u00b1}\\infty [\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042459472\">Recall that for any base [latex]b>0,b\\ne 1,[\/latex] the function [latex]y={b}^{x}[\/latex] is an exponential function with domain [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] and range [latex](0,\\infty ).[\/latex] If [latex]b>1,y={b}^{x}[\/latex] is increasing over [latex]`(\\text{\u2212}\\infty ,\\infty ).[\/latex] If [latex]0<b<1,[\/latex] [latex]y={b}^{x}[\/latex] is decreasing over [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] For the natural exponential function [latex]f(x)={e}^{x},[\/latex] [latex]e\\approx 2.718>1.[\/latex] Therefore, [latex]f(x)={e}^{x}[\/latex] is increasing on [latex]`(\\text{\u2212}\\infty ,\\infty )[\/latex] and the range is [latex]`(0,\\infty ).[\/latex] The exponential function [latex]f(x)={e}^{x}[\/latex] approaches [latex]\\infty[\/latex] as [latex]x\\to \\infty[\/latex] and approaches 0 as [latex]x\\to \\text{\u2212}\\infty[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#fs-id1165042542966\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_021\">(Figure)<\/a>.<\/p>\n<table id=\"fs-id1165042542966\" class=\"column-header\" summary=\"The table has two rows and six columns. The first column is a header column and it reads x and ex. After the header, the first row reads \u22125, \u22122, 0, 2, and 5. The second row reads 0.00674, 0.135, 1, 7.389, and 148.413.\">\n<caption>End behavior of the natural exponential function<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>-5<\/td>\n<td>-2<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>5<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]{e}^{x}[\/latex]<\/strong><\/td>\n<td>0.00674<\/td>\n<td>0.135<\/td>\n<td>1<\/td>\n<td>7.389<\/td>\n<td>148.413<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"CNX_Calc_Figure_04_06_021\" class=\"wp-caption aligncenter\">\n<div style=\"width: 277px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211126\/CNX_Calc_Figure_04_06_013.jpg\" alt=\"The function f(x) = ex is graphed.\" width=\"267\" height=\"234\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 21.<\/strong> The exponential function approaches zero as [latex]x\\to \\text{\u2212}\\infty [\/latex] and approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043218124\">Recall that the natural logarithm function [latex]f(x)=\\text{ln}(x)[\/latex] is the inverse of the natural exponential function [latex]y={e}^{x}.[\/latex] Therefore, the domain of [latex]f(x)=\\text{ln}(x)[\/latex] is [latex](0,\\infty )[\/latex] and the range is [latex](\\text{\u2212}\\infty ,\\infty ).[\/latex] The graph of [latex]f(x)=\\text{ln}(x)[\/latex] is the reflection of the graph of [latex]y={e}^{x}[\/latex] about the line [latex]y=x.[\/latex] Therefore, [latex]\\text{ln}(x)\\to \\text{\u2212}\\infty[\/latex] as [latex]x\\to {0}^{+}[\/latex] and [latex]\\text{ln}(x)\\to \\infty[\/latex] as [latex]x\\to \\infty[\/latex] as shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_022\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165042460463\">(Figure)<\/a>.<\/p>\n<table id=\"fs-id1165042460463\" class=\"column-header\" summary=\"The table has two rows and six columns. The first column is a header column and it reads x and ln(x). After the header, the first row reads 0.01, 0.1, 1, 10, and 100. The second row reads \u22124.605, \u22122.303, 0, 2.303, and 4.605.\">\n<caption>End behavior of the natural logarithm function<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>0.01<\/td>\n<td>0.1<\/td>\n<td>1<\/td>\n<td>10<\/td>\n<td>100<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]\\text{ln}(x)[\/latex]<\/strong><\/td>\n<td>-4.605<\/td>\n<td>-2.303<\/td>\n<td>0<\/td>\n<td>2.303<\/td>\n<td>4.605<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"CNX_Calc_Figure_04_06_022\" class=\"wp-caption aligncenter\">\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211128\/CNX_Calc_Figure_04_06_014.jpg\" alt=\"The function f(x) = ln(x) is graphed.\" width=\"417\" height=\"272\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 22.<\/strong> The natural logarithm function approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042469818\" class=\"textbox examples\">\n<h3>Determining End Behavior for a Transcendental Function<\/h3>\n<div id=\"fs-id1165042469820\" class=\"exercise\">\n<div id=\"fs-id1165042469822\" class=\"textbox\">\n<p id=\"fs-id1165042469828\">Find the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \\text{\u2212}\\infty[\/latex] for [latex]f(x)=\\frac{(2+3{e}^{x})}{(7-5{e}^{x})}[\/latex] and describe the end behavior of [latex]f.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042711624\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042711624\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711624\">To find the limit as [latex]x\\to \\infty ,[\/latex] divide the numerator and denominator by [latex]{e}^{x}\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1165042711652\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\underset{x\\to \\infty }{\\text{lim}}f(x)& =\\underset{x\\to \\infty }{\\text{lim}}\\frac{2+3{e}^{x}}{7-5{e}^{x}}\\hfill \\\\ & =\\underset{x\\to \\infty }{\\text{lim}}\\frac{(2\\text{\/}{e}^{x})+3}{(7\\text{\/}{e}^{x})-5}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165042499478\">As shown in <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_06_021\">(Figure)<\/a>, [latex]{e}^{x}\\to \\infty[\/latex] as [latex]x\\to \\infty .[\/latex] Therefore,<\/p>\n<div id=\"fs-id1165042499514\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{2}{{e}^{x}}=0=\\underset{x\\to \\infty }{\\text{lim}}\\frac{7}{{e}^{x}}.[\/latex]<\/div>\n<p id=\"fs-id1165042499577\">We conclude that [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=-\\frac{3}{5},[\/latex] and the graph of [latex]f[\/latex] approaches the horizontal asymptote [latex]y=-\\frac{3}{5}[\/latex] as [latex]x\\to \\infty .[\/latex] To find the limit as [latex]x\\to \\text{\u2212}\\infty ,[\/latex] use the fact that [latex]{e}^{x}\\to 0[\/latex] as [latex]x\\to \\text{\u2212}\\infty[\/latex] to conclude that [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\frac{2}{7},[\/latex] and therefore the graph of approaches the horizontal asymptote [latex]y=\\frac{2}{7}[\/latex] as [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042711306\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042711310\" class=\"exercise\">\n<div id=\"fs-id1165042711312\" class=\"textbox\">\n<p id=\"fs-id1165042711314\">Find the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \\text{\u2212}\\infty[\/latex] for [latex]f(x)=\\frac{(3{e}^{x}-4)}{(5{e}^{x}+2)}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042711402\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042711402\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711402\">[latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\frac{3}{5},[\/latex][latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=-2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042711473\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042711480\">[latex]\\underset{x\\to \\infty }{\\text{lim}}{e}^{x}=\\infty[\/latex] and [latex]\\underset{x\\to \\infty }{\\text{lim}}{e}^{x}=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042602932\" class=\"bc-section section\">\n<h1>Guidelines for Drawing the Graph of a Function<\/h1>\n<p id=\"fs-id1165042602938\">We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let\u2019s look at a general strategy to use when graphing any function.<\/p>\n<div id=\"fs-id1165042602944\" class=\"textbox key-takeaways problem-solving\">\n<h3>Problem-Solving Strategy: Drawing the Graph of a Function<\/h3>\n<p id=\"fs-id1165042602951\">Given a function [latex]f,[\/latex] use the following steps to sketch a graph of [latex]f\\text{:}[\/latex]<\/p>\n<ol id=\"fs-id1165042602969\">\n<li>Determine the domain of the function.<\/li>\n<li>Locate the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-intercepts.<\/li>\n<li>Evaluate [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)[\/latex] to determine the end behavior. If either of these limits is a finite number [latex]L,[\/latex] then [latex]y=L[\/latex] is a horizontal asymptote. If either of these limits is [latex]\\infty[\/latex] or [latex]\\text{\u2212}\\infty ,[\/latex] determine whether [latex]f[\/latex] has an oblique asymptote. If [latex]f[\/latex] is a rational function such that [latex]f(x)=\\frac{p(x)}{q(x)},[\/latex] where the degree of the numerator is greater than the degree of the denominator, then [latex]f[\/latex] can be written as\n<div id=\"fs-id1165042617521\" class=\"equation unnumbered\">[latex]f(x)=\\frac{p(x)}{q(x)}=g(x)+\\frac{r(x)}{q(x)},[\/latex]<\/div>\n<p>where the degree of [latex]r(x)[\/latex] is less than the degree of [latex]q(x).[\/latex] The values of [latex]f(x)[\/latex] approach the values of [latex]g(x)[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] If [latex]g(x)[\/latex] is a linear function, it is known as an <em>oblique asymptote<\/em>.<\/li>\n<li>Determine whether [latex]f[\/latex] has any vertical asymptotes.<\/li>\n<li>Calculate [latex]{f}^{\\prime }.[\/latex] Find all critical points and determine the intervals where [latex]f[\/latex] is increasing and where [latex]f[\/latex] is decreasing. Determine whether [latex]f[\/latex] has any local extrema.<\/li>\n<li>Calculate [latex]f\\text{\u2033}.[\/latex] Determine the intervals where [latex]f[\/latex] is concave up and where [latex]f[\/latex] is concave down. Use this information to determine whether [latex]f[\/latex] has any inflection points. The second derivative can also be used as an alternate means to determine or verify that [latex]f[\/latex] has a local extremum at a critical point.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1165042617762\">Now let\u2019s use this strategy to graph several different functions. We start by graphing a polynomial function.<\/p>\n<div id=\"fs-id1165042617768\" class=\"textbox examples\">\n<h3>Sketching a Graph of a Polynomial<\/h3>\n<div id=\"fs-id1165042617770\" class=\"exercise\">\n<div id=\"fs-id1165042617772\" class=\"textbox\">\n<p id=\"fs-id1165042707708\">Sketch a graph of [latex]f(x)={(x-1)}^{2}(x+2).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042707761\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042707761\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042707761\">Step 1. Since [latex]f[\/latex] is a polynomial, the domain is the set of all real numbers.<\/p>\n<p id=\"fs-id1165042707768\">Step 2. When [latex]x=0,f(x)=2.[\/latex] Therefore, the [latex]y[\/latex]-intercept is [latex](0,2).[\/latex] To find the [latex]x[\/latex]-intercepts, we need to solve the equation [latex]{(x-1)}^{2}(x+2)=0,[\/latex] gives us the [latex]x[\/latex]-intercepts [latex](1,0)[\/latex] and [latex](-2,0)[\/latex]<\/p>\n<p id=\"fs-id1165042707901\">Step 3. We need to evaluate the end behavior of [latex]f.[\/latex] As [latex]x\\to \\infty ,[\/latex] [latex]{(x-1)}^{2}\\to \\infty[\/latex] and [latex](x+2)\\to \\infty .[\/latex] Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\infty .[\/latex] As [latex]x\\to \\text{\u2212}\\infty ,[\/latex] [latex]{(x-1)}^{2}\\to \\infty[\/latex] and [latex](x+2)\\to \\text{\u2212}\\infty .[\/latex] Therefore, [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty .[\/latex] To get even more information about the end behavior of [latex]f,[\/latex] we can multiply the factors of [latex]f.[\/latex] When doing so, we see that<\/p>\n<div id=\"fs-id1165042525463\" class=\"equation unnumbered\">[latex]f(x)={(x-1)}^{2}(x+2)={x}^{3}-3x+2.[\/latex]<\/div>\n<p id=\"fs-id1165042525532\">Since the leading term of [latex]f[\/latex] is [latex]{x}^{3},[\/latex] we conclude that [latex]f[\/latex] behaves like [latex]y={x}^{3}[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex]<\/p>\n<p id=\"fs-id1165043382921\">Step 4. Since [latex]f[\/latex] is a polynomial function, it does not have any vertical asymptotes.<\/p>\n<p id=\"fs-id1165043382928\">Step 5. The first derivative of [latex]f[\/latex] is<\/p>\n<div class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=3{x}^{2}-3.[\/latex]<\/div>\n<p id=\"fs-id1165043382970\">Therefore, [latex]f[\/latex] has two critical points: [latex]x=1,-1.[\/latex] Divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the three smaller intervals: [latex](\\text{\u2212}\\infty ,-1),[\/latex] [latex](-1,1),[\/latex] and [latex](1,\\infty ).[\/latex] Then, choose test points [latex]x=-2,[\/latex] [latex]x=0,[\/latex] and [latex]x=2[\/latex] from these intervals and evaluate the sign of [latex]{f}^{\\prime }(x)[\/latex] at each of these test points, as shown in the following table.<\/p>\n<table id=\"fs-id1165043383127\" class=\"unnumbered\" summary=\"This table has four rows and four columns. The first row is a header row, and it reads Interval, Test Point, Sign of Derivative f\u2019(x) = 3x2 \u2013 3 = 3(x \u2013 1)(x + 1), and Conclusion. Under the header row, the first column reads (\u2212\u221e, \u22121), (\u22121, 1), and (1, \u221e). The second column reads x = \u22122, x = 0, and x = 2. The third column reads (+)(\u2212)(\u2212) = +, (+)(\u2212)(+) = \u2212, and (+)(+)(+) = +. The fourth column reads f is increasing, f is decreasing, and f is increasing.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of Derivative [latex]f\\prime (x)=3{x}^{2}-3=3(x-1)(x+1)[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\\text{\u2212}\\infty ,-1)[\/latex]<\/td>\n<td>[latex]x=-2[\/latex]<\/td>\n<td>[latex](\\text{+})(\\text{\u2212})(\\text{\u2212})=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](-1,1)[\/latex]<\/td>\n<td>[latex]x=0[\/latex]<\/td>\n<td>[latex](\\text{+})(\\text{\u2212})(\\text{+})=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](1,\\infty )[\/latex]<\/td>\n<td>[latex]x=2[\/latex]<\/td>\n<td>[latex](\\text{+})(\\text{+})(\\text{+})=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042539204\">From the table, we see that [latex]f[\/latex] has a local maximum at [latex]x=-1[\/latex] and a local minimum at [latex]x=1.[\/latex] Evaluating [latex]f(x)[\/latex] at those two points, we find that the local maximum value is [latex]f(-1)=4[\/latex] and the local minimum value is [latex]f(1)=0.[\/latex]<\/p>\n<p id=\"fs-id1165042539284\">Step 6. The second derivative of [latex]f[\/latex] is<\/p>\n<div id=\"fs-id1165042539292\" class=\"equation unnumbered\">[latex]f\\text{\u2033}(x)=6x.[\/latex]<\/div>\n<p id=\"fs-id1165042539318\">The second derivative is zero at [latex]x=0.[\/latex] Therefore, to determine the concavity of [latex]f,[\/latex] divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the smaller intervals [latex](\\text{\u2212}\\infty ,0)[\/latex] and [latex](0,\\infty ),[\/latex] and choose test points [latex]x=-1[\/latex] and [latex]x=1[\/latex] to determine the concavity of [latex]f[\/latex] on each of these smaller intervals as shown in the following table.<\/p>\n<table id=\"fs-id1165042381927\" class=\"unnumbered\" summary=\"This table has three rows and four columns. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019\u2019(x) = 6x, and Conclusion. Under the header row, the first column reads (\u2212\u221e, 0) and (0, \u221e). The second column reads x = \u22121 and x = 1. The third column reads \u2212 and +. The fourth column reads f is concave down and f is concave up.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]f\\text{\u2033}(x)=6x[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\\text{\u2212}\\infty ,0)[\/latex]<\/td>\n<td>[latex]x=-1[\/latex]<\/td>\n<td>[latex]-[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave down.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](0,\\infty )[\/latex]<\/td>\n<td>[latex]x=1[\/latex]<\/td>\n<td>[latex]+[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave up.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042382153\">We note that the information in the preceding table confirms the fact, found in step 5, that [latex]f[\/latex] has a local maximum at [latex]x=-1[\/latex] and a local minimum at [latex]x=1.[\/latex] In addition, the information found in step 5\u2014namely, [latex]f[\/latex] has a local maximum at [latex]x=-1[\/latex] and a local minimum at [latex]x=1,[\/latex] and [latex]{f}^{\\prime }(x)=0[\/latex] at those points\u2014combined with the fact that [latex]f\\text{\u2033}[\/latex] changes sign only at [latex]x=0[\/latex] confirms the results found in step 6 on the concavity of [latex]f.[\/latex]<\/p>\n<p id=\"fs-id1165042709981\">Combining this information, we arrive at the graph of [latex]f(x)={(x-1)}^{2}(x+2)[\/latex] shown in the following graph.<\/p>\n<p><span id=\"fs-id1165042710031\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211132\/CNX_Calc_Figure_04_06_015.jpg\" alt=\"The function f(x) = (x \u22121)2 (x + 2) is graphed. It crosses the x axis at x = \u22122 and touches the x axis at x = 1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042710046\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042710050\" class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1165042710055\">Sketch a graph of [latex]f(x)={(x-1)}^{3}(x+2).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042710106\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042710106\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165042710110\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211136\/CNX_Calc_Figure_04_06_028.jpg\" alt=\"The function f(x) = (x \u22121)3(x + 2) is graphed.\" \/><\/span><\/div>\n<div id=\"fs-id1165042710122\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042710130\">[latex]f[\/latex] is a fourth-degree polynomial.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042710140\" class=\"textbox examples\">\n<h3>Sketching a Rational Function<\/h3>\n<div id=\"fs-id1165042710142\" class=\"exercise\">\n<div id=\"fs-id1165042710144\" class=\"textbox\">\n<p id=\"fs-id1165042710150\">Sketch the graph of [latex]f(x)=\\frac{{x}^{2}}{(1-{x}^{2})}\\text{.}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042407414\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042407414\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042407414\">Step 1. The function [latex]f[\/latex] is defined as long as the denominator is not zero. Therefore, the domain is the set of all real numbers [latex]x[\/latex] except [latex]x=\\text{\u00b1}1.[\/latex]<\/p>\n<p id=\"fs-id1165042407440\">Step 2. Find the intercepts. If [latex]x=0,[\/latex] then [latex]f(x)=0,[\/latex] so 0 is an intercept. If [latex]y=0,[\/latex] then [latex]\\frac{{x}^{2}}{(1-{x}^{2})}=0,[\/latex] which implies [latex]x=0.[\/latex] Therefore, [latex](0,0)[\/latex] is the only intercept.<\/p>\n<p id=\"fs-id1165042407553\">Step 3. Evaluate the limits at infinity. Since [latex]f[\/latex] is a rational function, divide the numerator and denominator by the highest power in the denominator: [latex]{x}^{2}.[\/latex] We obtain<\/p>\n<div id=\"fs-id1165042407571\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}\\frac{1}{\\frac{1}{{x}^{2}}-1}=-1.[\/latex]<\/div>\n<p id=\"fs-id1165042407655\">Therefore, [latex]f[\/latex] has a horizontal asymptote of [latex]y=-1[\/latex] as [latex]x\\to \\infty[\/latex] and [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\n<p id=\"fs-id1165042491004\">Step 4. To determine whether [latex]f[\/latex] has any vertical asymptotes, first check to see whether the denominator has any zeroes. We find the denominator is zero when [latex]x=\\text{\u00b1}1.[\/latex] To determine whether the lines [latex]x=1[\/latex] or [latex]x=-1[\/latex] are vertical asymptotes of [latex]f,[\/latex] evaluate [latex]\\underset{x\\to 1}{\\text{lim}}f(x)[\/latex] and [latex]\\underset{x\\to \\text{\u2212}1}{\\text{lim}}f(x).[\/latex] By looking at each one-sided limit as [latex]x\\to 1,[\/latex] we see that<\/p>\n<div id=\"fs-id1165042491126\" class=\"equation unnumbered\">[latex]\\underset{x\\to {1}^{+}}{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\text{\u2212}\\infty \\text{ and }\\underset{x\\to {1}^{-}}{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\infty .[\/latex]<\/div>\n<p id=\"fs-id1165042491222\">In addition, by looking at each one-sided limit as [latex]x\\to \\text{\u2212}1,[\/latex] we find that<\/p>\n<div id=\"fs-id1165042491242\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\text{\u2212}{1}^{+}}{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\infty \\text{ and }\\underset{x\\to \\text{\u2212}{1}^{-}}{\\text{lim}}\\frac{{x}^{2}}{1-{x}^{2}}=\\text{\u2212}\\infty .[\/latex]<\/div>\n<p id=\"fs-id1165043262266\">Step 5. Calculate the first derivative:<\/p>\n<div id=\"fs-id1165043262270\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{(1-{x}^{2})(2x)-{x}^{2}(-2x)}{{(1-{x}^{2})}^{2}}=\\frac{2x}{{(1-{x}^{2})}^{2}}.[\/latex]<\/div>\n<p id=\"fs-id1165043262390\">Critical points occur at points [latex]x[\/latex] where [latex]{f}^{\\prime }(x)=0[\/latex] or [latex]{f}^{\\prime }(x)[\/latex] is undefined. We see that [latex]{f}^{\\prime }(x)=0[\/latex] when [latex]x=0.[\/latex] The derivative [latex]{f}^{\\prime }[\/latex] is not undefined at any point in the domain of [latex]f.[\/latex] However, [latex]x=\\text{\u00b1}1[\/latex] are not in the domain of [latex]f.[\/latex] Therefore, to determine where [latex]f[\/latex] is increasing and where [latex]f[\/latex] is decreasing, divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into four smaller intervals: [latex](\\text{\u2212}\\infty ,-1),[\/latex] [latex](-1,0),[\/latex] [latex](0,1),[\/latex] and [latex](1,\\infty ),[\/latex] and choose a test point in each interval to determine the sign of [latex]{f}^{\\prime }(x)[\/latex] in each of these intervals. The values [latex]x=-2,[\/latex] [latex]x=-\\frac{1}{2},[\/latex] [latex]x=\\frac{1}{2},[\/latex] and [latex]x=2[\/latex] are good choices for test points as shown in the following table.<\/p>\n<table id=\"fs-id1165043341618\" class=\"unnumbered\" summary=\"This table has four columns and five rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019(x) = 2x\/(1 \u2212 x2)2, and Conclusion. Under the header row, the first column reads (\u2212\u221e, \u22121), (\u22121, 0), (0, 1), and (1, \u221e). The second column reads x = \u22122, x = \u22121\/2, x = 1\/2, and x = 2. The third column reads \u2212\/+ = \u2212, \u2212\/+ = \u2212, +\/+ = +, and +\/+ = +. The fourth column reads f is decreasing, f is decreasing, f is increasing, and f is increasing.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]{f}^{\\prime }(x)=\\frac{2x}{{(1-{x}^{2})}^{2}}[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\\text{\u2212}\\infty ,-1)[\/latex]<\/td>\n<td>[latex]x=-2[\/latex]<\/td>\n<td>[latex]\\text{\u2212}\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](-1,0)[\/latex]<\/td>\n<td>[latex]x=-1\\text{\/}2[\/latex]<\/td>\n<td>[latex]\\text{\u2212}\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](0,1)[\/latex]<\/td>\n<td>[latex]x=1\\text{\/}2[\/latex]<\/td>\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](1,\\infty )[\/latex]<\/td>\n<td>[latex]x=2[\/latex]<\/td>\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165043183348\">From this analysis, we conclude that [latex]f[\/latex] has a local minimum at [latex]x=0[\/latex] but no local maximum.<\/p>\n<p id=\"fs-id1165043183355\">Step 6. Calculate the second derivative:<\/p>\n<div id=\"fs-id1165043183358\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill f\\text{\u2033}(x)& \\hfill =\\frac{{(1-{x}^{2})}^{2}(2)-2x(2(1-{x}^{2})(-2x))}{{(1-{x}^{2})}^{4}}\\\\ & =\\frac{(1-{x}^{2})\\left[2(1-{x}^{2})+8{x}^{2}\\right]}{{(1-{x}^{2})}^{4}}\\hfill \\\\ & =\\frac{2(1-{x}^{2})+8{x}^{2}}{{(1-{x}^{2})}^{3}}\\hfill \\\\ & =\\frac{6{x}^{2}+2}{{(1-{x}^{2})}^{3}}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165043384128\">To determine the intervals where [latex]f[\/latex] is concave up and where [latex]f[\/latex] is concave down, we first need to find all points [latex]x[\/latex] where [latex]f\\text{\u2033}(x)=0[\/latex] or [latex]f\\text{\u2033}(x)[\/latex] is undefined. Since the numerator [latex]6{x}^{2}+2\\ne 0[\/latex] for any [latex]x,[\/latex] [latex]f\\text{\u2033}(x)[\/latex] is never zero. Furthermore, [latex]f\\text{\u2033}[\/latex] is not undefined for any [latex]x[\/latex] in the domain of [latex]f.[\/latex] However, as discussed earlier, [latex]x=\\text{\u00b1}1[\/latex] are not in the domain of [latex]f.[\/latex] Therefore, to determine the concavity of [latex]f,[\/latex] we divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the three smaller intervals [latex](\\text{\u2212}\\infty ,-1),[\/latex] [latex](-1,-1),[\/latex] and [latex](1,\\infty ),[\/latex] and choose a test point in each of these intervals to evaluate the sign of [latex]f\\text{\u2033}(x).[\/latex] in each of these intervals. The values [latex]x=-2,[\/latex] [latex]x=0,[\/latex] and [latex]x=2[\/latex] are possible test points as shown in the following table.<\/p>\n<table id=\"fs-id1165043384406\" class=\"unnumbered\" summary=\"This table has four columns and four rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019\u2019(x) = (6x2 + 2)\/(1 \u2212 x2)3, and Conclusion. Under the header row, the first column reads (\u2212\u221e, \u22121), (\u22121, 1), and (1, \u221e). The second column reads x = \u22122, x = 0, and x = 2. The third column reads +\/\u2212 = \u2212, +\/+ = +, and +\/\u2212 = \u2212. The fourth column reads f is concave down, f is concave up, and f is concave down.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]f\\text{\u2033}(x)=\\frac{6{x}^{2}+2}{{(1-{x}^{2})}^{3}}[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\\text{\u2212}\\infty ,-1)[\/latex]<\/td>\n<td>[latex]x=-2[\/latex]<\/td>\n<td>[latex]+\\text{\/}-=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave down.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](-1,-1)[\/latex]<\/td>\n<td>[latex]x=0[\/latex]<\/td>\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave up.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](1,\\infty )[\/latex]<\/td>\n<td>[latex]x=2[\/latex]<\/td>\n<td>[latex]+\\text{\/}-=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave down.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042592688\">Combining all this information, we arrive at the graph of [latex]f[\/latex] shown below. Note that, although [latex]f[\/latex] changes concavity at [latex]x=-1[\/latex] and [latex]x=1,[\/latex] there are no inflection points at either of these places because [latex]f[\/latex] is not continuous at [latex]x=-1[\/latex] or [latex]x=1.[\/latex]<\/p>\n<p><span id=\"fs-id1165042592749\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211139\/CNX_Calc_Figure_04_06_016.jpg\" alt=\"The function f(x) = x2\/(1 \u2212 x2) is graphed. It has asymptotes y = \u22121, x = \u22121, and x = 1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042592764\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042592768\" class=\"exercise\">\n<div id=\"fs-id1165042592770\" class=\"textbox\">\n<p id=\"fs-id1165042592772\">Sketch a graph of [latex]f(x)=\\frac{(3x+5)}{(8+4x)}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042592825\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042592825\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165042592830\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211142\/CNX_Calc_Figure_04_06_029.jpg\" alt=\"The function f(x) = (3x + 5)\/(8 + 4x) is graphed. It appears to have asymptotes at x = \u22122 and y = 1.\" \/><\/span><\/div>\n<div id=\"fs-id1165042592842\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042592850\">A line [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex] if the limit as [latex]x\\to \\infty[\/latex] or the limit as [latex]x\\to \\text{\u2212}\\infty[\/latex] of [latex]f(x)[\/latex] is [latex]L.[\/latex] A line [latex]x=a[\/latex] is a vertical asymptote if at least one of the one-sided limits of [latex]f[\/latex] as [latex]x\\to a[\/latex] is [latex]\\infty[\/latex] or [latex]\\text{\u2212}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042592955\" class=\"textbox examples\">\n<h3>Sketching a Rational Function with an Oblique Asymptote<\/h3>\n<div id=\"fs-id1165042592957\" class=\"exercise\">\n<div id=\"fs-id1165042592959\" class=\"textbox\">\n<p id=\"fs-id1165042592964\">Sketch the graph of [latex]f(x)=\\frac{{x}^{2}}{(x-1)}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042593006\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042593006\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042593006\">Step 1. The domain of [latex]f[\/latex] is the set of all real numbers [latex]x[\/latex] except [latex]x=1.[\/latex]<\/p>\n<p id=\"fs-id1165042712534\">Step 2. Find the intercepts. We can see that when [latex]x=0,[\/latex] [latex]f(x)=0,[\/latex] so [latex](0,0)[\/latex] is the only intercept.<\/p>\n<p id=\"fs-id1165042712584\">Step 3. Evaluate the limits at infinity. Since the degree of the numerator is one more than the degree of the denominator, [latex]f[\/latex] must have an oblique asymptote. To find the oblique asymptote, use long division of polynomials to write<\/p>\n<div id=\"fs-id1165042712594\" class=\"equation unnumbered\">[latex]f(x)=\\frac{{x}^{2}}{x-1}=x+1+\\frac{1}{x-1}.[\/latex]<\/div>\n<p id=\"fs-id1165042712649\">Since [latex]1\\text{\/}(x-1)\\to 0[\/latex] as [latex]x\\to \\text{\u00b1}\\infty ,[\/latex] [latex]f(x)[\/latex] approaches the line [latex]y=x+1[\/latex] as [latex]x\\to \\text{\u00b1}\\infty .[\/latex] The line [latex]y=x+1[\/latex] is an oblique asymptote for [latex]f.[\/latex]<\/p>\n<p id=\"fs-id1165042712758\">Step 4. To check for vertical asymptotes, look at where the denominator is zero. Here the denominator is zero at [latex]x=1.[\/latex] Looking at both one-sided limits as [latex]x\\to 1,[\/latex] we find<\/p>\n<div id=\"fs-id1165042712789\" class=\"equation unnumbered\">[latex]\\underset{x\\to {1}^{+}}{\\text{lim}}\\frac{{x}^{2}}{x-1}=\\infty \\text{ and }\\underset{x\\to {1}^{-}}{\\text{lim}}\\frac{{x}^{2}}{x-1}=\\text{\u2212}\\infty .[\/latex]<\/div>\n<p id=\"fs-id1165042403315\">Therefore, [latex]x=1[\/latex] is a vertical asymptote, and we have determined the behavior of [latex]f[\/latex] as [latex]x[\/latex] approaches 1 from the right and the left.<\/p>\n<p id=\"fs-id1165042403340\">Step 5. Calculate the first derivative:<\/p>\n<div id=\"fs-id1165042403344\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{(x-1)(2x)-{x}^{2}(1)}{{(x-1)}^{2}}=\\frac{{x}^{2}-2x}{{(x-1)}^{2}}.[\/latex]<\/div>\n<p id=\"fs-id1165042403459\">We have [latex]{f}^{\\prime }(x)=0[\/latex] when [latex]{x}^{2}-2x=x(x-2)=0.[\/latex] Therefore, [latex]x=0[\/latex] and [latex]x=2[\/latex] are critical points. Since [latex]f[\/latex] is undefined at [latex]x=1,[\/latex] we need to divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the smaller intervals [latex](\\text{\u2212}\\infty ,0),[\/latex] [latex](0,1),[\/latex] [latex](1,2),[\/latex] and [latex](2,\\infty ),[\/latex] and choose a test point from each interval to evaluate the sign of [latex]{f}^{\\prime }(x)[\/latex] in each of these smaller intervals. For example, let [latex]x=-1,[\/latex] [latex]x=\\frac{1}{2},[\/latex] [latex]x=\\frac{3}{2},[\/latex] and [latex]x=3[\/latex] be the test points as shown in the following table.<\/p>\n<table id=\"fs-id1165042710566\" class=\"unnumbered\" summary=\"This table has four columns and five rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019(x) = (x2 \u2212 2x)\/(x \u2212 1)2 = x(x \u2212 2)\/(x \u2212 1)2, and Conclusion. Under the header row, the first column reads (\u2212\u221e, 0), (0, 1), (1, 2), and (2, \u221e). The second column reads x = \u22121, x = 1\/2, x = 3\/2, and x = 3. The third column reads (\u2212)(\u2212)\/+ = +, (+)(\u2212)\/+ = \u2212, (+)(\u2212)\/+ = \u2212, and (+)(+)\/+ = +. The fourth column reads f is increasing, f is decreasing, f is decreasing, and f is increasing.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]f\\prime (x)=\\frac{{x}^{2}-2x}{{(x-1)}^{2}}=\\frac{x(x-2)}{{(x-1)}^{2}}[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\\text{\u2212}\\infty ,0)[\/latex]<\/td>\n<td>[latex]x=-1[\/latex]<\/td>\n<td>[latex](\\text{\u2212})(\\text{\u2212})\\text{\/}+=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](0,1)[\/latex]<\/td>\n<td>[latex]x=1\\text{\/}2[\/latex]<\/td>\n<td>[latex](\\text{+})(\\text{\u2212})\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](1,2)[\/latex]<\/td>\n<td>[latex]x=3\\text{\/}2[\/latex]<\/td>\n<td>[latex](\\text{+})(\\text{\u2212})\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](2,\\infty )[\/latex]<\/td>\n<td>[latex]x=3[\/latex]<\/td>\n<td>[latex](\\text{+})(\\text{+})\\text{\/}+=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042476453\">From this table, we see that [latex]f[\/latex] has a local maximum at [latex]x=0[\/latex] and a local minimum at [latex]x=2.[\/latex] The value of [latex]f[\/latex] at the local maximum is [latex]f(0)=0[\/latex] and the value of [latex]f[\/latex] at the local minimum is [latex]f(2)=4.[\/latex] Therefore, [latex](0,0)[\/latex] and [latex](2,4)[\/latex] are important points on the graph.<\/p>\n<p id=\"fs-id1165042464546\">Step 6. Calculate the second derivative:<\/p>\n<div id=\"fs-id1165042464549\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill f\\text{\u2033}(x)& =\\frac{{(x-1)}^{2}(2x-2)-({x}^{2}-2x)(2(x-1))}{{(x-1)}^{4}}\\hfill \\\\ & =\\frac{(x-1)\\left[(x-1)(2x-2)-2({x}^{2}-2x)\\right]}{{(x-1)}^{4}}\\hfill \\\\ & =\\frac{(x-1)(2x-2)-2({x}^{2}-2x)}{{(x-1)}^{3}}\\hfill \\\\ & =\\frac{2{x}^{2}-4x+2-(2{x}^{2}-4x)}{{(x-1)}^{3}}\\hfill \\\\ & =\\frac{2}{{(x-1)}^{3}}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165042652400\">We see that [latex]f\\text{\u2033}(x)[\/latex] is never zero or undefined for [latex]x[\/latex] in the domain of [latex]f.[\/latex] Since [latex]f[\/latex] is undefined at [latex]x=1,[\/latex] to check concavity we just divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the two smaller intervals [latex](\\text{\u2212}\\infty ,1)[\/latex] and [latex](1,\\infty ),[\/latex] and choose a test point from each interval to evaluate the sign of [latex]f\\text{\u2033}(x)[\/latex] in each of these intervals. The values [latex]x=0[\/latex] and [latex]x=2[\/latex] are possible test points as shown in the following table.<\/p>\n<table id=\"fs-id1165042652542\" class=\"unnumbered\" summary=\"This table has four columns and three rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019\u2019(x) = 2\/(x \u2212 1)3, and Conclusion. Under the header row, the first column reads (\u2212\u221e, 1) and (1, \u221e). The column row reads x = 0 and x = 2. The third column reads +\/\u2212 = \u2212 and +\/+ = +. The fourth column reads f is concave down and f is concave up.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]f\\text{\u2033}(x)=\\frac{2}{{(x-1)}^{3}}[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\\text{\u2212}\\infty ,1)[\/latex]<\/td>\n<td>[latex]x=0[\/latex]<\/td>\n<td>[latex]+\\text{\/}-=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave down.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](1,\\infty )[\/latex]<\/td>\n<td>[latex]x=2[\/latex]<\/td>\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave up.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042606958\">From the information gathered, we arrive at the following graph for [latex]f.[\/latex]<\/p>\n<p><span id=\"fs-id1165042606968\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211145\/CNX_Calc_Figure_04_06_017.jpg\" alt=\"The function f(x) = x2\/(x \u2212 1) is graphed. It has asymptotes y = x + 1 and x = 1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042606982\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042606986\" class=\"exercise\">\n<div id=\"fs-id1165042606988\" class=\"textbox\">\n<p id=\"fs-id1165042606991\">Find the oblique asymptote for [latex]f(x)=\\frac{(3{x}^{3}-2x+1)}{(2{x}^{2}-4)}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042607059\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042607059\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042607059\">[latex]y=\\frac{3}{2}x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042607076\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042607084\">Use long division of polynomials.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042607090\" class=\"textbox examples\">\n<h3>Sketching the Graph of a Function with a Cusp<\/h3>\n<div id=\"fs-id1165042607093\" class=\"exercise\">\n<div id=\"fs-id1165042607095\" class=\"textbox\">\n<p id=\"fs-id1165042607100\">Sketch a graph of [latex]f(x)={(x-1)}^{2\\text{\/}3}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042607145\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042607145\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042607145\">Step 1. Since the cube-root function is defined for all real numbers [latex]x[\/latex] and [latex]{(x-1)}^{2\\text{\/}3}={(\\sqrt[3]{x-1})}^{2},[\/latex] the domain of [latex]f[\/latex] is all real numbers.<\/p>\n<p id=\"fs-id1165042607208\">Step 2: To find the [latex]y[\/latex]-intercept, evaluate [latex]f(0).[\/latex] Since [latex]f(0)=1,[\/latex] the [latex]y[\/latex]-intercept is [latex](0,1).[\/latex] To find the [latex]x[\/latex]-intercept, solve [latex]{(x-1)}^{2\\text{\/}3}=0.[\/latex] The solution of this equation is [latex]x=1,[\/latex] so the [latex]x[\/latex]-intercept is [latex](1,0).[\/latex]<\/p>\n<p id=\"fs-id1165042583208\">Step 3: Since [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}{(x-1)}^{2\\text{\/}3}=\\infty ,[\/latex] the function continues to grow without bound as [latex]x\\to \\infty[\/latex] and [latex]x\\to \\text{\u2212}\\infty .[\/latex]<\/p>\n<p id=\"fs-id1165042583285\">Step 4: The function has no vertical asymptotes.<\/p>\n<p id=\"fs-id1165042583288\">Step 5: To determine where [latex]f[\/latex] is increasing or decreasing, calculate [latex]{f}^{\\prime }.[\/latex] We find<\/p>\n<div id=\"fs-id1165042583307\" class=\"equation unnumbered\">[latex]{f}^{\\prime }(x)=\\frac{2}{3}{(x-1)}^{-1\\text{\/}3}=\\frac{2}{3{(x-1)}^{1\\text{\/}3}}.[\/latex]<\/div>\n<p id=\"fs-id1165042583388\">This function is not zero anywhere, but it is undefined when [latex]x=1.[\/latex] Therefore, the only critical point is [latex]x=1.[\/latex] Divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the smaller intervals [latex](\\text{\u2212}\\infty ,1)[\/latex] and [latex](1,\\infty ),[\/latex] and choose test points in each of these intervals to determine the sign of [latex]{f}^{\\prime }(x)[\/latex] in each of these smaller intervals. Let [latex]x=0[\/latex] and [latex]x=2[\/latex] be the test points as shown in the following table.<\/p>\n<table id=\"fs-id1165042504467\" class=\"unnumbered\" summary=\"This table has four columns and three rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019(x) = 2\/(3(x \u2212 1)1\/3), and Conclusion. Under the header row, the first column reads (\u2212\u221e, 1) and (1, \u221e). The second column reads x = 0 and x = 2. The third column reads +\/\u2212 = \u2212 and +\/+ = +. The fourth column reads f is decreasing and f is increasing.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]{f}^{\\prime }(x)=\\frac{2}{3{(x-1)}^{1\\text{\/}3}}[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\\text{\u2212}\\infty ,1)[\/latex]<\/td>\n<td>[latex]x=0[\/latex]<\/td>\n<td>[latex]+\\text{\/}-=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](1,\\infty )[\/latex]<\/td>\n<td>[latex]x=2[\/latex]<\/td>\n<td>[latex]+\\text{\/}+=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042504734\">We conclude that [latex]f[\/latex] has a local minimum at [latex]x=1.[\/latex] Evaluating [latex]f[\/latex] at [latex]x=1,[\/latex] we find that the value of [latex]f[\/latex] at the local minimum is zero. Note that [latex]{f}^{\\prime }(1)[\/latex] is undefined, so to determine the behavior of the function at this critical point, we need to examine [latex]\\underset{x\\to 1}{\\text{lim}}{f}^{\\prime }(x).[\/latex] Looking at the one-sided limits, we have<\/p>\n<div id=\"fs-id1165042510168\" class=\"equation unnumbered\">[latex]\\underset{x\\to {1}^{+}}{\\text{lim}}\\frac{2}{3{(x-1)}^{1\\text{\/}3}}=\\infty \\text{ and }\\underset{x\\to {1}^{-}}{\\text{lim}}\\frac{2}{3{(x-1)}^{1\\text{\/}3}}=\\text{\u2212}\\infty .[\/latex]<\/div>\n<p id=\"fs-id1165042510285\">Therefore, [latex]f[\/latex] has a cusp at [latex]x=1.[\/latex]<\/p>\n<p id=\"fs-id1165042510304\">Step 6: To determine concavity, we calculate the second derivative of [latex]f\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1165042510313\" class=\"equation unnumbered\">[latex]f\\text{\u2033}(x)=-\\frac{2}{9}{(x-1)}^{-4\\text{\/}3}=\\frac{-2}{9{(x-1)}^{4\\text{\/}3}}.[\/latex]<\/div>\n<p id=\"fs-id1165042510398\">We find that [latex]f\\text{\u2033}(x)[\/latex] is defined for all [latex]x,[\/latex] but is undefined when [latex]x=1.[\/latex] Therefore, divide the interval [latex](\\text{\u2212}\\infty ,\\infty )[\/latex] into the smaller intervals [latex](\\text{\u2212}\\infty ,1)[\/latex] and [latex](1,\\infty ),[\/latex] and choose test points to evaluate the sign of [latex]f\\text{\u2033}(x)[\/latex] in each of these intervals. As we did earlier, let [latex]x=0[\/latex] and [latex]x=2[\/latex] be test points as shown in the following table.<\/p>\n<table id=\"fs-id1165042510530\" class=\"unnumbered\" summary=\"This table has four columns and three rows. The first row is a header row, and it reads Interval, Test Point, Sign of f\u2019\u2019(x) = \u22122\/(9(x \u2212 1)4\/3), and Conclusion. Under the header row, the first column reads (\u2212\u221e, 1) and (1, \u221e). The second column reads x = 0 and x = 2. The third column reads \u2212\/+ = \u2212 and \u2212\/+ = \u2212. The fourth column reads f is concave down and f is concave down.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]f\\text{\u2033}(x)=\\frac{-2}{9{(x-1)}^{4\\text{\/}3}}[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\\text{\u2212}\\infty ,1)[\/latex]<\/td>\n<td>[latex]x=0[\/latex]<\/td>\n<td>[latex]\\text{\u2212}\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave down.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](1,\\infty )[\/latex]<\/td>\n<td>[latex]x=2[\/latex]<\/td>\n<td>[latex]\\text{\u2212}\\text{\/}+=\\text{\u2212}[\/latex]<\/td>\n<td>[latex]f[\/latex] is concave down.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042643969\">From this table, we conclude that [latex]f[\/latex] is concave down everywhere. Combining all of this information, we arrive at the following graph for [latex]f.[\/latex]<\/p>\n<p><span id=\"fs-id1165042643991\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211148\/CNX_Calc_Figure_04_06_018.jpg\" alt=\"The function f(x) = (x \u2212 1)2\/3 is graphed. It touches the x axis at x = 1, where it comes to something of a sharp point and then flairs out on either side.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042644007\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042644011\" class=\"exercise\">\n<div id=\"fs-id1165042644013\" class=\"textbox\">\n<p id=\"fs-id1165042644015\">Consider the function [latex]f(x)=5-{x}^{2\\text{\/}3}.[\/latex] Determine the point on the graph where a cusp is located. Determine the end behavior of [latex]f.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042644060\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042644060\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042644060\">The function [latex]f[\/latex] has a cusp at [latex](0,5)[\/latex] [latex]\\underset{x\\to {0}^{-}}{\\text{lim}}{f}^{\\prime }(x)=\\infty ,[\/latex] [latex]\\underset{x\\to {0}^{+}}{\\text{lim}}{f}^{\\prime }(x)=\\text{\u2212}\\infty .[\/latex] For end behavior, [latex]\\underset{x\\to \\text{\u00b1}\\infty }{\\text{lim}}f(x)=\\text{\u2212}\\infty .[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042633566\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042633573\">A function [latex]f[\/latex] has a cusp at a point [latex]a[\/latex] if [latex]f(a)[\/latex] exists, [latex]f\\prime (a)[\/latex] is undefined, one of the one-sided limits as [latex]x\\to a[\/latex] of [latex]f\\prime (x)[\/latex] is [latex]+\\infty ,[\/latex] and the other one-sided limit is [latex]\\text{\u2212}\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042633662\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165042633670\">\n<li>The limit of [latex]f(x)[\/latex] is [latex]L[\/latex] as [latex]x\\to \\infty[\/latex] (or as [latex]x\\to \\text{\u2212}\\infty )[\/latex] if the values [latex]f(x)[\/latex] become arbitrarily close to [latex]L[\/latex] as [latex]x[\/latex] becomes sufficiently large.<\/li>\n<li>The limit of [latex]f(x)[\/latex] is [latex]\\infty[\/latex] as [latex]x\\to \\infty[\/latex] if [latex]f(x)[\/latex] becomes arbitrarily large as [latex]x[\/latex] becomes sufficiently large. The limit of [latex]f(x)[\/latex] is [latex]\\text{\u2212}\\infty[\/latex] as [latex]x\\to \\infty[\/latex] if [latex]f(x)<0[\/latex] and [latex]|f(x)|[\/latex] becomes arbitrarily large as [latex]x[\/latex] becomes sufficiently large. We can define the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]\\text{\u2212}\\infty[\/latex] similarly.<\/li>\n<li>For a polynomial function [latex]p(x)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\\text{\u2026}+{a}_{1}x+{a}_{0},[\/latex] where [latex]{a}_{n}\\ne 0,[\/latex] the end behavior is determined by the leading term [latex]{a}_{n}{x}^{n}.[\/latex] If [latex]n\\ne 0,[\/latex] [latex]p(x)[\/latex] approaches [latex]\\infty[\/latex] or [latex]\\text{\u2212}\\infty[\/latex] at each end.<\/li>\n<li>For a rational function [latex]f(x)=\\frac{p(x)}{q(x)},[\/latex] the end behavior is determined by the relationship between the degree of [latex]p[\/latex] and the degree of [latex]q.[\/latex] If the degree of [latex]p[\/latex] is less than the degree of [latex]q,[\/latex] the line [latex]y=0[\/latex] is a horizontal asymptote for [latex]f.[\/latex] If the degree of [latex]p[\/latex] is equal to the degree of [latex]q,[\/latex] then the line [latex]y=\\frac{{a}_{n}}{{b}_{n}}[\/latex] is a horizontal asymptote, where [latex]{a}_{n}[\/latex] and [latex]{b}_{n}[\/latex] are the leading coefficients of [latex]p[\/latex] and [latex]q,[\/latex] respectively. If the degree of [latex]p[\/latex] is greater than the degree of [latex]q,[\/latex] then [latex]f[\/latex] approaches [latex]\\infty[\/latex] or [latex]\\text{\u2212}\\infty[\/latex] at each end.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165042640624\" class=\"textbox exercises\">\n<p id=\"fs-id1165042640628\">For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.<\/p>\n<div id=\"fs-id1165042640632\" class=\"exercise\">\n<div id=\"fs-id1165042640635\" class=\"textbox\"><span id=\"fs-id1165042640637\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211151\/CNX_Calc_Figure_04_06_201.jpg\" alt=\"The function graphed decreases very rapidly as it approaches x = 1 from the left, and on the other side of x = 1, it seems to start near infinity and then decrease rapidly.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042640653\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042640653\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042640653\">[latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042640666\" class=\"exercise\">\n<div id=\"fs-id1165042640668\" class=\"textbox\"><span id=\"fs-id1165042640674\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211153\/CNX_Calc_Figure_04_06_202.jpg\" alt=\"The function graphed increases very rapidly as it approaches x = \u22123 from the left, and on the other side of x = \u22123, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165043197220\" class=\"exercise\">\n<div id=\"fs-id1165043197222\" class=\"textbox\"><span id=\"fs-id1165043197228\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211156\/CNX_Calc_Figure_04_06_203.jpg\" alt=\"The function graphed decreases very rapidly as it approaches x = \u22121 from the left, and on the other side of x = \u22121, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043197243\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043197243\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043197243\">[latex]x=-1,x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043197265\" class=\"exercise\">\n<div id=\"fs-id1165043197267\" class=\"textbox\"><span id=\"fs-id1165043197269\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211158\/CNX_Calc_Figure_04_06_204.jpg\" alt=\"The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side of the U being at x = 1. On the other side of x = 1, there is another U shape pointing down, with its other side being at x = 2. On the other side of x = 2, the graph seems to start near negative infinity and then increase rapidly.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165043197318\" class=\"exercise\">\n<div id=\"fs-id1165043197320\" class=\"textbox\"><span id=\"fs-id1165043197322\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211201\/CNX_Calc_Figure_04_06_205.jpg\" alt=\"The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side being a normal function that appears as if it will take the entirety of the values of the x-axis.\" \/><\/span><\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043197340\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043197340\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043197340\">[latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043197353\">For the following functions [latex]f(x),[\/latex] determine whether there is an asymptote at [latex]x=a.[\/latex] Justify your answer without graphing on a calculator.<\/p>\n<div id=\"fs-id1165043197383\" class=\"exercise\">\n<div id=\"fs-id1165043197385\" class=\"textbox\">\n<p id=\"fs-id1165043197388\">[latex]f(x)=\\frac{x+1}{{x}^{2}+5x+4},a=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043197447\" class=\"exercise\">\n<div id=\"fs-id1165043197449\" class=\"textbox\">\n<p id=\"fs-id1165043197451\">[latex]f(x)=\\frac{x}{x-2},a=2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043197490\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043197490\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043197490\">Yes, there is a vertical asymptote<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043197495\" class=\"exercise\">\n<div id=\"fs-id1165043197497\" class=\"textbox\">\n<p id=\"fs-id1165043197500\">[latex]f(x)={(x+2)}^{3\\text{\/}2},a=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043197555\" class=\"exercise\">\n<div id=\"fs-id1165043197557\" class=\"textbox\">\n<p id=\"fs-id1165043197560\">[latex]f(x)={(x-1)}^{-1\\text{\/}3},a=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043197610\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043197610\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043197610\">Yes, there is vertical asymptote<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043197616\" class=\"exercise\">\n<div id=\"fs-id1165043197618\" class=\"textbox\">\n<p id=\"fs-id1165043197620\">[latex]f(x)=1+{x}^{-2\\text{\/}5},a=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043377731\">For the following exercises, evaluate the limit.<\/p>\n<div id=\"fs-id1165043377734\" class=\"exercise\">\n<div id=\"fs-id1165043377736\" class=\"textbox\">\n<p id=\"fs-id1165043377738\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{1}{3x+6}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043377772\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043377772\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043377772\">0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043377780\" class=\"exercise\">\n<div id=\"fs-id1165043377782\" class=\"textbox\">\n<p id=\"fs-id1165043377784\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{2x-5}{4x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043377833\" class=\"exercise\">\n<div id=\"fs-id1165043377835\" class=\"textbox\">\n<p id=\"fs-id1165043377837\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{{x}^{2}-2x+5}{x+2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043377884\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043377884\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043377884\">[latex]\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043377891\" class=\"exercise\">\n<div id=\"fs-id1165043377893\" class=\"textbox\">\n<p id=\"fs-id1165043377896\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{3{x}^{3}-2x}{{x}^{2}+2x+8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043377963\" class=\"exercise\">\n<div id=\"fs-id1165043377965\" class=\"textbox\">\n<p id=\"fs-id1165043377967\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{{x}^{4}-4{x}^{3}+1}{2-2{x}^{2}-7{x}^{4}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043378034\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043378034\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043378034\">[latex]-\\frac{1}{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043378048\" class=\"exercise\">\n<div id=\"fs-id1165043378050\" class=\"textbox\">\n<p id=\"fs-id1165043378052\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{3x}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043378100\" class=\"exercise\">\n<div id=\"fs-id1165043216345\" class=\"textbox\">\n<p id=\"fs-id1165043216347\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{\\sqrt{4{x}^{2}-1}}{x+2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043216394\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043216394\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043216394\">-2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043216402\" class=\"exercise\">\n<div id=\"fs-id1165043216404\" class=\"textbox\">\n<p id=\"fs-id1165043216407\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{4x}{\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043216454\" class=\"exercise\">\n<div id=\"fs-id1165043216457\" class=\"textbox\">\n<p id=\"fs-id1165043216459\">[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{4x}{\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043216501\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043216501\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043216501\">-4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043216510\" class=\"exercise\">\n<div id=\"fs-id1165043216512\" class=\"textbox\">\n<p id=\"fs-id1165043216514\">[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{2\\sqrt{x}}{x-\\sqrt{x}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043216563\">For the following exercises, find the horizontal and vertical asymptotes.<\/p>\n<div id=\"fs-id1165043216566\" class=\"exercise\">\n<div id=\"fs-id1165043216569\" class=\"textbox\">\n<p id=\"fs-id1165043216571\">[latex]f(x)=x-\\frac{9}{x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043216600\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043216600\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043216600\">Horizontal: none, vertical: [latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043216614\" class=\"exercise\">\n<div id=\"fs-id1165043216616\" class=\"textbox\">\n<p id=\"fs-id1165043216618\">[latex]f(x)=\\frac{1}{1-{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043216679\" class=\"exercise\">\n<div id=\"fs-id1165043216681\" class=\"textbox\">\n<p id=\"fs-id1165043216683\">[latex]f(x)=\\frac{{x}^{3}}{4-{x}^{2}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043216720\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043216720\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043216720\">Horizontal: none, vertical: [latex]x=\\text{\u00b1}2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043216737\" class=\"exercise\">\n<div id=\"fs-id1165043216739\" class=\"textbox\">\n<p id=\"fs-id1165043216741\">[latex]f(x)=\\frac{{x}^{2}+3}{{x}^{2}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043210294\" class=\"exercise\">\n<div id=\"fs-id1165043210296\" class=\"textbox\">\n<p id=\"fs-id1165043210298\">[latex]f(x)= \\sin (x) \\sin (2x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043210340\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043210340\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043210340\">Horizontal: none, vertical: none<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043210346\" class=\"exercise\">\n<div id=\"fs-id1165043210348\" class=\"textbox\">\n<p id=\"fs-id1165043210350\">[latex]f(x)= \\cos x+ \\cos (3x)+ \\cos (5x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043210409\" class=\"exercise\">\n<div id=\"fs-id1165043210411\" class=\"textbox\">\n<p id=\"fs-id1165043210413\">[latex]f(x)=\\frac{x \\sin (x)}{{x}^{2}-1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043210457\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043210457\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043210457\">Horizontal: [latex]y=0,[\/latex] vertical: [latex]x=\\text{\u00b1}1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043210485\" class=\"exercise\">\n<div id=\"fs-id1165043210487\" class=\"textbox\">\n<p id=\"fs-id1165043210489\">[latex]f(x)=\\frac{x}{ \\sin (x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043210550\" class=\"exercise\">\n<div id=\"fs-id1165043210552\" class=\"textbox\">\n<p id=\"fs-id1165043210554\">[latex]f(x)=\\frac{1}{{x}^{3}+{x}^{2}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043210591\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043210591\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043210591\">Horizontal: [latex]y=0,[\/latex] vertical: [latex]x=0[\/latex] and [latex]x=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043210626\" class=\"exercise\">\n<div id=\"fs-id1165043210628\" class=\"textbox\">\n<p id=\"fs-id1165043210630\">[latex]f(x)=\\frac{1}{x-1}-2x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043349198\" class=\"exercise\">\n<div id=\"fs-id1165043349200\" class=\"textbox\">\n<p id=\"fs-id1165043349202\">[latex]f(x)=\\frac{{x}^{3}+1}{{x}^{3}-1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043349244\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043349244\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043349244\">Horizontal: [latex]y=1,[\/latex] vertical: [latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043349269\" class=\"exercise\">\n<div id=\"fs-id1165043349272\" class=\"textbox\">\n<p id=\"fs-id1165043349274\">[latex]f(x)=\\frac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043349346\" class=\"exercise\">\n<div id=\"fs-id1165043349348\" class=\"textbox\">\n<p id=\"fs-id1165043349350\">[latex]f(x)=x- \\sin x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043349378\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043349378\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043349378\">Horizontal: none, vertical: none<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043349383\" class=\"exercise\">\n<div id=\"fs-id1165043349385\" class=\"textbox\">\n<p id=\"fs-id1165043349388\">[latex]f(x)=\\frac{1}{x}-\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043349432\">For the following exercises, construct a function [latex]f(x)[\/latex] that has the given asymptotes.<\/p>\n<div id=\"fs-id1165043349448\" class=\"exercise\">\n<div id=\"fs-id1165043349450\" class=\"textbox\">\n<p id=\"fs-id1165043349452\">[latex]x=1[\/latex] and [latex]y=2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043349476\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043349476\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043349476\">Answers will vary, for example: [latex]y=\\frac{2x}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043349502\" class=\"exercise\">\n<div id=\"fs-id1165043349504\" class=\"textbox\">\n<p id=\"fs-id1165043349506\">[latex]x=1[\/latex] and [latex]y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043349559\" class=\"exercise\">\n<div id=\"fs-id1165043349561\" class=\"textbox\">\n<p id=\"fs-id1165043349563\">[latex]y=4,[\/latex][latex]x=-1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043349589\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043349589\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043349589\">Answers will vary, for example: [latex]y=\\frac{4x}{x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043349615\" class=\"exercise\">\n<div id=\"fs-id1165043349617\" class=\"textbox\">\n<p id=\"fs-id1165043349619\">[latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042709113\">For the following exercises, graph the function on a graphing calculator on the window [latex]x=\\left[-5,5\\right][\/latex] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.<\/p>\n<div id=\"fs-id1165042709138\" class=\"exercise\">\n<div id=\"fs-id1165042709140\" class=\"textbox\">\n<p id=\"fs-id1165042709142\"><strong>[T]<\/strong>[latex]f(x)=\\frac{1}{x+10}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042709177\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042709177\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042709177\">[latex]y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042709190\" class=\"exercise\">\n<div id=\"fs-id1165042709192\" class=\"textbox\">\n<p id=\"fs-id1165042709194\"><strong>[T]<\/strong>[latex]f(x)=\\frac{x+1}{{x}^{2}+7x+6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042709257\" class=\"exercise\">\n<div id=\"fs-id1165042709259\" class=\"textbox\">\n<p id=\"fs-id1165042709261\"><strong>[T]<\/strong>[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}{x}^{2}+10x+25[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042709306\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042709306\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042709306\">[latex]\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042709313\" class=\"exercise\">\n<div id=\"fs-id1165042709316\" class=\"textbox\">\n<p id=\"fs-id1165042709318\"><strong>[T]<\/strong>[latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}\\frac{x+2}{{x}^{2}+7x+6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042709385\" class=\"exercise\">\n<div id=\"fs-id1165042709387\" class=\"textbox\">\n<p id=\"fs-id1165042709389\"><strong>[T]<\/strong>[latex]\\underset{x\\to \\infty }{\\text{lim}}\\frac{3x+2}{x+5}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042709433\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042709433\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042709433\">[latex]y=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042709446\">For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.<\/p>\n<div id=\"fs-id1165042709451\" class=\"exercise\">\n<div id=\"fs-id1165042709453\" class=\"textbox\">\n<p id=\"fs-id1165042709456\">[latex]y=3{x}^{2}+2x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042709501\" class=\"exercise\">\n<div id=\"fs-id1165042709504\" class=\"textbox\">\n<p id=\"fs-id1165042709506\">[latex]y={x}^{3}-3{x}^{2}+4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042558120\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042558120\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165042558126\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211203\/CNX_Calc_Figure_04_06_207.jpg\" alt=\"The function starts in the third quadrant, increases to pass through (\u22121, 0), increases to a maximum and y intercept at 4, decreases to touch (2, 0), and then increases to (4, 20).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042558140\" class=\"exercise\">\n<div id=\"fs-id1165042558142\" class=\"textbox\">\n<p id=\"fs-id1165042558144\">[latex]y=\\frac{2x+1}{{x}^{2}+6x+5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042558205\" class=\"exercise\">\n<div id=\"fs-id1165042558207\" class=\"textbox\">\n<p id=\"fs-id1165042558209\">[latex]y=\\frac{{x}^{3}+4{x}^{2}+3x}{3x+9}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042558253\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042558253\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165042558258\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211205\/CNX_Calc_Figure_04_06_209.jpg\" alt=\"An upward-facing parabola with minimum between x = 0 and x = \u22121 with y intercept between 0 and 1.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042558272\" class=\"exercise\">\n<div id=\"fs-id1165042558274\" class=\"textbox\">\n<p id=\"fs-id1165042558276\">[latex]y=\\frac{{x}^{2}+x-2}{{x}^{2}-3x-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042558343\" class=\"exercise\">\n<div id=\"fs-id1165042558345\" class=\"textbox\">\n<p id=\"fs-id1165042558347\">[latex]y=\\sqrt{{x}^{2}-5x+4}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042558376\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042558376\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165042558381\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211208\/CNX_Calc_Figure_04_06_211.jpg\" alt=\"This graph starts at (\u22122, 4) and decreases in a convex way to (1, 0). Then the graph starts again at (4, 0) and increases in a convex way to (6, 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042558395\" class=\"exercise\">\n<div id=\"fs-id1165042558397\" class=\"textbox\">\n<p id=\"fs-id1165042558399\">[latex]y=2x\\sqrt{16-{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042558446\" class=\"exercise\">\n<div id=\"fs-id1165042558448\" class=\"textbox\">\n<p id=\"fs-id1165042558450\">[latex]y=\\frac{ \\cos x}{x},[\/latex] on [latex]x=\\left[-2\\pi ,2\\pi \\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042558496\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042558496\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165042558507\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211210\/CNX_Calc_Figure_04_06_213.jpg\" alt=\"This graph has vertical asymptote at x = 0. The first part of the function occurs in the second and third quadrants and starts in the third quadrant just below (\u22122\u03c0, 0), increases and passes through the x axis at \u22123\u03c0\/2, reaches a maximum and then decreases through the x axis at \u2212\u03c0\/2 before approaching the asymptote. On the other side of the asymptote, the function starts in the first quadrant, decreases quickly to pass through \u03c0\/2, decreases to a local minimum and then increases through (3\u03c0\/2, 0) before staying just above (2\u03c0, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042558519\" class=\"exercise\">\n<div id=\"fs-id1165042558521\" class=\"textbox\">\n<p id=\"fs-id1165042558523\">[latex]y={e}^{x}-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043404024\" class=\"exercise\">\n<div id=\"fs-id1165043404026\" class=\"textbox\">\n<p id=\"fs-id1165043404029\">[latex]y=x \\tan x,x=\\left[\\text{\u2212}\\pi ,\\pi \\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043404067\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043404067\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165043404074\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211213\/CNX_Calc_Figure_04_06_215.jpg\" alt=\"This graph has vertical asymptotes at x = \u00b1\u03c0\/2. The graph is symmetric about the y axis, so describing the left hand side will be sufficient. The function starts at (\u2212\u03c0, 0) and decreases quickly to the asymptote. Then it starts on the other side of the asymptote in the second quadrant and decreases to the the origin.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043404088\" class=\"exercise\">\n<div id=\"fs-id1165043404090\" class=\"textbox\">\n<p id=\"fs-id1165043404092\">[latex]y=x\\text{ln}(x),x>0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043404141\" class=\"exercise\">\n<div id=\"fs-id1165043404143\" class=\"textbox\">\n<p id=\"fs-id1165043404145\">[latex]y={x}^{2} \\sin (x),x=\\left[-2\\pi ,2\\pi \\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043404194\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043404194\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165043404201\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211215\/CNX_Calc_Figure_04_06_217.jpg\" alt=\"This function starts at (\u22122\u03c0, 0), increases to near (\u22123\u03c0\/2, 25), decreases through (\u2212\u03c0, 0), achieves a local minimum and then increases through the origin. On the other side of the origin, the graph is the same but flipped, that is, it is congruent to the other half by a rotation of 180 degrees.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043404215\" class=\"exercise\">\n<div id=\"fs-id1165043404217\" class=\"textbox\">\n<p id=\"fs-id1165043404219\">For [latex]f(x)=\\frac{P(x)}{Q(x)}[\/latex] to have an asymptote at [latex]y=2[\/latex] then the polynomials [latex]P(x)[\/latex] and [latex]Q(x)[\/latex] must have what relation?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043404361\" class=\"exercise\">\n<div id=\"fs-id1165043404363\" class=\"textbox\">\n<p id=\"fs-id1165043404366\">For [latex]f(x)=\\frac{P(x)}{Q(x)}[\/latex] to have an asymptote at [latex]x=0,[\/latex] then the polynomials [latex]P(x)[\/latex] and [latex]Q(x).[\/latex] must have what relation?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043208543\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043208543\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043208543\">[latex]Q(x).[\/latex] must have have [latex]{x}^{k+1}[\/latex] as a factor, where [latex]P(x)[\/latex] has [latex]{x}^{k}[\/latex] as a factor.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043208598\" class=\"exercise\">\n<div id=\"fs-id1165043208600\" class=\"textbox\">\n<p id=\"fs-id1165043208602\">If [latex]{f}^{\\prime }(x)[\/latex] has asymptotes at [latex]y=3[\/latex] and [latex]x=1,[\/latex] then [latex]f(x)[\/latex] has what asymptotes?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043208676\" class=\"exercise\">\n<div id=\"fs-id1165043208678\" class=\"textbox\">\n<p id=\"fs-id1165043208681\">Both [latex]f(x)=\\frac{1}{(x-1)}[\/latex] and [latex]g(x)=\\frac{1}{{(x-1)}^{2}}[\/latex] have asymptotes at [latex]x=1[\/latex] and [latex]y=0.[\/latex] What is the most obvious difference between these two functions?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043208777\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043208777\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043208777\">[latex]\\underset{x\\to {1}^{-}}{\\text{lim}}f(x)\\text{ and }\\underset{x\\to {1}^{-}}{\\text{lim}}g(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043208845\" class=\"exercise\">\n<div id=\"fs-id1165043208847\" class=\"textbox\">\n<p id=\"fs-id1165043208849\">True or false: Every ratio of polynomials has vertical asymptotes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165043208865\" class=\"definition\">\n<dt>end behavior<\/dt>\n<dd id=\"fs-id1165043208870\">the behavior of a function as [latex]x\\to \\infty[\/latex] and [latex]x\\to \\text{\u2212}\\infty[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043208899\" class=\"definition\">\n<dt>horizontal asymptote<\/dt>\n<dd id=\"fs-id1165043208905\">if [latex]\\underset{x\\to \\infty }{\\text{lim}}f(x)=L[\/latex] or [latex]\\underset{x\\to \\text{\u2212}\\infty }{\\text{lim}}f(x)=L,[\/latex] then [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042462524\" class=\"definition\">\n<dt>infinite limit at infinity<\/dt>\n<dd id=\"fs-id1165042462530\">a function that becomes arbitrarily large as [latex]x[\/latex] becomes large<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042462539\" class=\"definition\">\n<dt>limit at infinity<\/dt>\n<dd id=\"fs-id1165042462545\">the limiting value, if it exists, of a function as [latex]x\\to \\infty[\/latex] or [latex]x\\to \\text{\u2212}\\infty[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042462574\" class=\"definition\">\n<dt>oblique asymptote<\/dt>\n<dd id=\"fs-id1165042462579\">the line [latex]y=mx+b[\/latex] if [latex]f(x)[\/latex] approaches it as [latex]x\\to \\infty[\/latex] or [latex]x\\to \\text{\u2212}\\infty[\/latex]<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":311,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1991","chapter","type-chapter","status-publish","hentry"],"part":1878,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1991","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1991\/revisions"}],"predecessor-version":[{"id":2523,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1991\/revisions\/2523"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/1878"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1991\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=1991"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=1991"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=1991"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=1991"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}