{"id":407,"date":"2021-02-04T02:01:27","date_gmt":"2021-02-04T02:01:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=407"},"modified":"2022-03-16T05:47:06","modified_gmt":"2022-03-16T05:47:06","slug":"limits-at-infinity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/limits-at-infinity\/","title":{"raw":"Limits at Infinity","rendered":"Limits at Infinity"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Calculate the limit of a function as \ud835\udc65 increases or decreases without bound<\/li>\r\n \t<li>Recognize a horizontal asymptote on the graph of a function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165043145047\" class=\"bc-section section\">\r\n\r\nWe 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>. We have\u00a0looked at vertical asymptotes in other modules; in this section, we deal with horizontal and oblique asymptotes.\r\n<h3>Limits at Infinity and Horizontal Asymptotes<\/h3>\r\n<p id=\"fs-id1165043107285\">Recall that [latex]\\underset{x \\to a}{\\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 Figure 1 and numerically in the table beneath it, 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 }{\\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]\u2212\\infty [\/latex] of [latex]f(x)[\/latex] is 2 and write [latex]\\underset{x\\to a}{\\lim}f(x)=2[\/latex].<\/p>\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\/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\" \/> Figure 1. The function approaches the asymptote [latex]y=2[\/latex] as [latex]x[\/latex] approaches [latex]\\pm \\infty[\/latex].[\/caption]\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 \\pm \\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 \\infty}{\\lim}f(x)=L[\/latex]. Similarly, we say the limit as [latex]x\\to \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 \u2212\\infty }{\\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 shaded\">\r\n<div class=\"title\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=L[\/latex]<\/div>\r\n&nbsp;\r\n\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 \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] (Figure 2). For example, for the function [latex]f(x)=\\frac{1}{x}[\/latex], since [latex]\\underset{x\\to \\infty }{\\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 shaded\">\r\n<div class=\"title\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165042973921\">If [latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex] or [latex]\\underset{x \\to \u2212\\infty}{\\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[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\" \/> Figure 2. (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 \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]\r\n<p id=\"fs-id1165042647732\">A function cannot cross a vertical asymptote because the graph must approach infinity (or negative infinity) 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 Figure 3 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[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\" \/> Figure 3. The graph of [latex]f(x)=\\cos x\/x+1[\/latex] crosses its horizontal asymptote [latex]y=1[\/latex] an infinite number of times.[\/caption]\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\/calculus1\/chapter\/why-it-matters-limits\/\">Why It Matters: 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=\"textbook exercises\">\r\n<h3>Example: 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 }{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\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)=\\dfrac{\\sin x}{x}[\/latex]<\/li>\r\n \t<li>[latex]f(x)= \\tan^{-1} (x)[\/latex]<\/li>\r\n<\/ol>\r\n[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 }{\\lim}(5-\\frac{2}{x^2})=\\underset{x\\to \\infty }{\\lim}5-2(\\underset{x\\to \\infty }{\\lim}\\frac{1}{x})(\\underset{x\\to \\infty }{\\lim}\\frac{1}{x})=5-2 \\cdot 0=5[\/latex]. Similarly, [latex]\\underset{x\\to -\\infty }{\\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 \\pm \\infty [\/latex] as shown in the following graph.[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\" \/> Figure 4. This function approaches a horizontal asymptote as [latex]x\\to \\pm \\infty[\/latex].[\/caption]<\/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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{-1}{x}=0=\\underset{x\\to \\infty }{\\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\sin x}{x}=0[\/latex]<\/div>\r\nSimilarly,\r\n<div id=\"fs-id1165043122536\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty}{\\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 \\pm \\infty [\/latex] as shown in the following graph.\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\" \/> Figure 5. This function crosses its horizontal asymptote multiple times.[\/caption]<\/li>\r\n \t<li>To evaluate [latex]\\underset{x\\to \\infty }{\\lim} \\tan^{-1} (x)[\/latex] and [latex]\\underset{x\\to \u2212\\infty}{\\lim} \\tan^{-1} (x)[\/latex], we first consider the graph of [latex]y= \\tan (x)[\/latex] over the interval [latex](\u2212\\pi \/2,\\pi \/2)[\/latex] as shown in the following graph.[caption id=\"\" align=\"aligncenter\" width=\"492\"]<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.\" width=\"492\" height=\"347\" \/> Figure 6. The graph of [latex] \\tan x[\/latex] has vertical asymptotes at [latex]x=\\pm \\frac{\\pi }{2}[\/latex][\/caption]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165043092430\">Since<\/p>\r\n\r\n<div id=\"fs-id1165043119614\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to (\\pi\/2)^-}{\\lim} \\tan x=\\infty [\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042514177\">it follows that<\/p>\r\n\r\n<div id=\"fs-id1165042563973\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim} \\tan^{-1} (x)=\\frac{\\pi }{2}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043097156\">Similarly, since<\/p>\r\n\r\n<div id=\"fs-id1165042923290\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to (\\pi\/2)^+}{\\lim} \\tan x=\u2212\\infty[\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043131939\">it follows that<\/p>\r\n\r\n<div id=\"fs-id1165043056813\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty}{\\lim} \\tan^{-1} (x)=-\\frac{\\pi }{2}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nAs 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.\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\" \/> Figure 7. This function has two horizontal asymptotes.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Computing Limits at Infinity.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=70&amp;end=307&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.6LimitsAtInfinityAndAsymptotes70to307_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.6 Limits at Infinity and Asymptotes\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1165042320881\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165043315935\">Evaluate [latex]\\underset{x\\to \u2212\\infty}{\\lim}\\left(3+\\frac{4}{x}\\right)[\/latex] and [latex]\\underset{x\\to \\infty }{\\lim}\\left(3+\\frac{4}{x}\\right)[\/latex]. Determine the horizontal asymptotes of [latex]f(x)=3+\\frac{4}{x}[\/latex], if any.<\/p>\r\n[reveal-answer q=\"2473508\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"2473508\"]\r\n<p id=\"fs-id1165042318511\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}1\/x=0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n[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[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]169165[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042333169\" class=\"bc-section section\">\r\n<h3>Infinite Limits at Infinity<\/h3>\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 \u2212\\infty )[\/latex]. In this case, we write [latex]\\underset{x\\to \\infty }{\\lim}f(x)=\\infty [\/latex] (or [latex]\\underset{x\\to \u2212\\infty }{\\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 \u2212\\infty )[\/latex], we write [latex]\\underset{x\\to \\infty }{\\lim}f(x)=\u2212\\infty [\/latex] (or [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=\u2212\\infty )[\/latex].<\/p>\r\n<p id=\"fs-id1165042606820\">For example, consider the function [latex]f(x)=x^3[\/latex]. As seen in the table below and Figure 8, as [latex]x\\to \\infty [\/latex] the values [latex]f(x)[\/latex] become arbitrarily large. Therefore, [latex]\\underset{x\\to \\infty }{\\lim}x^3=\\infty[\/latex]. On the other hand, as [latex]x\\to \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 \u2212\\infty }{\\lim}x^3=\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 \\pm \\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>[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\" \/> Figure 8. For this function, the functional values approach infinity as [latex]x\\to \\pm \\infty[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1165043276353\" class=\"textbox shaded\">\r\n<div class=\"title\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=\\infty [\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=\u2212\\infty[\/latex]<\/div>\r\n&nbsp;\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 \u2212\\infty[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042328702\" class=\"bc-section section\">\r\n<h3>Formal Definitions<\/h3>\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 shaded\">\r\n<div class=\"title\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\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]\\varepsilon &gt;0[\/latex], there exists [latex]N&gt;0[\/latex] such that<\/p>\r\n\r\n<div id=\"fs-id1165043395062\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|f(x)-L|&lt;\\varepsilon [\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042512686\">(see Figure 9).<\/p>\r\n&nbsp;\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]\\varepsilon &gt;0[\/latex], there exists [latex]N&lt;0[\/latex] such that<\/p>\r\n\r\n<div id=\"fs-id1165042331766\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|f(x)-L|&lt;\\varepsilon [\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=L[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>[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\" \/> Figure 9. For a function with a limit at infinity, for all [latex]x&gt;N[\/latex], [latex]|f(x)-L|&lt;\\varepsilon [\/latex].[\/caption]<\/div>\r\n<p id=\"fs-id1165043396243\">Earlier in this section, we used graphical evidence and numerical evidence to conclude that [latex]\\underset{x\\to \\infty }{\\lim}\\left(2+\\frac{1}{x}\\right)=2[\/latex]. Here we use the formal definition of limit at infinity to prove this result rigorously.<\/p>\r\n\r\n<div class=\"textbook exercises\">\r\n<h3>Example: A Finite Limit at Infinity Example<\/h3>\r\n<p id=\"fs-id1165042587296\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}\\left(2+\\frac{1}{x}\\right)=2[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1165042369578\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042369578\"]\r\n<p id=\"fs-id1165042369578\">Let [latex]\\varepsilon &gt;0[\/latex]. Let [latex]N=\\frac{1}{\\varepsilon }[\/latex]. Therefore, for all [latex]x&gt;N[\/latex], we have<\/p>\r\n\r\n<div id=\"fs-id1165043312498\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left| 2+\\dfrac{1}{x}-2 \\right| =\\left| \\dfrac{1}{x} \\right|=\\dfrac{1}{x}&lt;\\dfrac{1}{N}=\\varepsilon[\/latex].[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: A Finite Limit at Infinity Example.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=450&amp;end=630&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.6LimitsAtInfinityAndAsymptotes450to630_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.6 Limits at Infinity and Asymptotes\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1165042480092\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165042480099\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}\\left(3 - \\dfrac{1}{x^2}\\right)=3[\/latex].<\/p>\r\n[reveal-answer q=\"9822541\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"9822541\"]\r\n<p id=\"fs-id1165042332065\">Let [latex]N=\\dfrac{1}{\\sqrt{\\varepsilon }}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165042367887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042367887\"]\r\n<p id=\"fs-id1165042367887\">Let [latex]\\varepsilon &gt;0[\/latex]. Let [latex]N=\\dfrac{1}{\\sqrt{\\varepsilon }}[\/latex]. Therefore, for all [latex]x&gt;N[\/latex], we have<\/p>\r\n<p id=\"fs-id1165042376362\">[latex]|3-\\dfrac{1}{x^2}-3|=\\dfrac{1}{x^2}&lt;\\dfrac{1}{N^2}=\\varepsilon [\/latex]<\/p>\r\n<p id=\"fs-id1165042320298\">Therefore, [latex]\\underset{x\\to \\infty }{\\lim}\\left(3-\\frac{1}{x^2}\\right)=3[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\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 shaded\">\r\n<div class=\"title\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=\\infty [\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]f(x)&gt;M[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042374733\">for all [latex]x&gt;N[\/latex] (see Figure 10).<\/p>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=\u2212\\infty [\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]f(x)&lt;M[\/latex]<\/div>\r\n&nbsp;\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 \u2212\\infty[\/latex].<\/p>\r\n\r\n<\/div>\r\n[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\" \/> Figure 10. For a function with an infinite limit at infinity, for all [latex]x&gt;N[\/latex], [latex]f(x)&gt;M[\/latex].[\/caption]\r\n<p id=\"fs-id1165042705963\">Earlier, we used graphical evidence (Figure 8) and numerical evidence (the table beneath it) to conclude that [latex]\\underset{x\\to \\infty }{\\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=\"textbook exercises\">\r\n<h3>Example: An Infinite Limit at Infinity<\/h3>\r\n<p id=\"fs-id1165043395589\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}x^3=\\infty[\/latex].<\/p>\r\n[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\" style=\"text-align: center;\">[latex]x^3&gt;N^3=(\\sqrt[3]{M})^3=M[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042604681\">Therefore, [latex]\\underset{x\\to \\infty }{\\lim}x^3=\\infty[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: An Infinite Limit at Infinity.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=666&amp;end=816&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266835\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266835\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.6LimitsAtInfinityAndAsymptotes666to816_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.6 Limits at Infinity and Asymptotes\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1165042323710\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165042320226\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}3x^2=\\infty[\/latex].<\/p>\r\n[reveal-answer q=\"80775166\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"80775166\"]\r\n<p id=\"fs-id1165043219104\">Let [latex]N=\\sqrt{\\frac{M}{3}}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[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]3x^2&gt;3N^2=3(\\sqrt{\\frac{M}{3}})^2=\\frac{3M}{3}=M[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]16109[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate the limit of a function as \ud835\udc65 increases or decreases without bound<\/li>\n<li>Recognize a horizontal asymptote on the graph of a function<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165043145047\" class=\"bc-section section\">\n<p>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>. We have\u00a0looked at vertical asymptotes in other modules; in this section, we deal with horizontal and oblique asymptotes.<\/p>\n<h3>Limits at Infinity and Horizontal Asymptotes<\/h3>\n<p id=\"fs-id1165043107285\">Recall that [latex]\\underset{x \\to a}{\\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 Figure 1 and numerically in the table beneath it, 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 }{\\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]\u2212\\infty[\/latex] of [latex]f(x)[\/latex] is 2 and write [latex]\\underset{x\\to a}{\\lim}f(x)=2[\/latex].<\/p>\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\">Figure 1. The function approaches the asymptote [latex]y=2[\/latex] as [latex]x[\/latex] approaches [latex]\\pm \\infty[\/latex].<\/p>\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 \\pm \\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 \\infty}{\\lim}f(x)=L[\/latex]. Similarly, we say the limit as [latex]x\\to \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 \u2212\\infty }{\\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 shaded\">\n<div class=\"title\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=L[\/latex]<\/div>\n<p>&nbsp;<\/p>\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 \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] (Figure 2). For example, for the function [latex]f(x)=\\frac{1}{x}[\/latex], since [latex]\\underset{x\\to \\infty }{\\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 shaded\">\n<div class=\"title\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<\/div>\n<p id=\"fs-id1165042973921\">If [latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex] or [latex]\\underset{x \\to \u2212\\infty}{\\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 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\">Figure 2. (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 \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<p id=\"fs-id1165042647732\">A function cannot cross a vertical asymptote because the graph must approach infinity (or negative infinity) 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 Figure 3 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 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\">Figure 3. The graph of [latex]f(x)=\\cos x\/x+1[\/latex] crosses its horizontal asymptote [latex]y=1[\/latex] an infinite number of times.<\/p>\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\/calculus1\/chapter\/why-it-matters-limits\/\">Why It Matters: 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=\"textbook exercises\">\n<h3>Example: 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 }{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\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)=\\dfrac{\\sin x}{x}[\/latex]<\/li>\n<li>[latex]f(x)= \\tan^{-1} (x)[\/latex]<\/li>\n<\/ol>\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 }{\\lim}(5-\\frac{2}{x^2})=\\underset{x\\to \\infty }{\\lim}5-2(\\underset{x\\to \\infty }{\\lim}\\frac{1}{x})(\\underset{x\\to \\infty }{\\lim}\\frac{1}{x})=5-2 \\cdot 0=5[\/latex]. Similarly, [latex]\\underset{x\\to -\\infty }{\\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 \\pm \\infty[\/latex] as shown in the following graph.\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\">Figure 4. This function approaches a horizontal asymptote as [latex]x\\to \\pm \\infty[\/latex].<\/p>\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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{-1}{x}=0=\\underset{x\\to \\infty }{\\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\sin x}{x}=0[\/latex]<\/div>\n<p>Similarly,<\/p>\n<div id=\"fs-id1165043122536\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty}{\\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 \\pm \\infty[\/latex] as shown in the following graph.<\/p>\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\">Figure 5. This function crosses its horizontal asymptote multiple times.<\/p>\n<\/div>\n<\/li>\n<li>To evaluate [latex]\\underset{x\\to \\infty }{\\lim} \\tan^{-1} (x)[\/latex] and [latex]\\underset{x\\to \u2212\\infty}{\\lim} \\tan^{-1} (x)[\/latex], we first consider the graph of [latex]y= \\tan (x)[\/latex] over the interval [latex](\u2212\\pi \/2,\\pi \/2)[\/latex] as shown in the following graph.\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\/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.\" width=\"492\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. The graph of [latex] \\tan x[\/latex] has vertical asymptotes at [latex]x=\\pm \\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165043092430\">Since<\/p>\n<div id=\"fs-id1165043119614\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to (\\pi\/2)^-}{\\lim} \\tan x=\\infty[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042514177\">it follows that<\/p>\n<div id=\"fs-id1165042563973\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim} \\tan^{-1} (x)=\\frac{\\pi }{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043097156\">Similarly, since<\/p>\n<div id=\"fs-id1165042923290\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to (\\pi\/2)^+}{\\lim} \\tan x=\u2212\\infty[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043131939\">it follows that<\/p>\n<div id=\"fs-id1165043056813\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty}{\\lim} \\tan^{-1} (x)=-\\frac{\\pi }{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>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 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\">Figure 7. This function has two horizontal asymptotes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Computing Limits at Infinity.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=70&amp;end=307&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.6LimitsAtInfinityAndAsymptotes70to307_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.6 Limits at Infinity and Asymptotes&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1165042320881\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165043315935\">Evaluate [latex]\\underset{x\\to \u2212\\infty}{\\lim}\\left(3+\\frac{4}{x}\\right)[\/latex] and [latex]\\underset{x\\to \\infty }{\\lim}\\left(3+\\frac{4}{x}\\right)[\/latex]. Determine the horizontal asymptotes of [latex]f(x)=3+\\frac{4}{x}[\/latex], if any.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q2473508\">Hint<\/span><\/p>\n<div id=\"q2473508\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042318511\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}1\/x=0[\/latex]<\/p>\n<\/div>\n<\/div>\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>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm169165\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=169165&theme=oea&iframe_resize_id=ohm169165&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165042333169\" class=\"bc-section section\">\n<h3>Infinite Limits at Infinity<\/h3>\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 \u2212\\infty )[\/latex]. In this case, we write [latex]\\underset{x\\to \\infty }{\\lim}f(x)=\\infty[\/latex] (or [latex]\\underset{x\\to \u2212\\infty }{\\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 \u2212\\infty )[\/latex], we write [latex]\\underset{x\\to \\infty }{\\lim}f(x)=\u2212\\infty[\/latex] (or [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=\u2212\\infty )[\/latex].<\/p>\n<p id=\"fs-id1165042606820\">For example, consider the function [latex]f(x)=x^3[\/latex]. As seen in the table below and Figure 8, as [latex]x\\to \\infty[\/latex] the values [latex]f(x)[\/latex] become arbitrarily large. Therefore, [latex]\\underset{x\\to \\infty }{\\lim}x^3=\\infty[\/latex]. On the other hand, as [latex]x\\to \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 \u2212\\infty }{\\lim}x^3=\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 \\pm \\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>\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\">Figure 8. For this function, the functional values approach infinity as [latex]x\\to \\pm \\infty[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043276353\" class=\"textbox shaded\">\n<div class=\"title\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=\\infty[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=\u2212\\infty[\/latex]<\/div>\n<p>&nbsp;<\/p>\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 \u2212\\infty[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042328702\" class=\"bc-section section\">\n<h3>Formal Definitions<\/h3>\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 shaded\">\n<div class=\"title\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\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]\\varepsilon >0[\/latex], there exists [latex]N>0[\/latex] such that<\/p>\n<div id=\"fs-id1165043395062\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|f(x)-L|<\\varepsilon[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042512686\">(see Figure 9).<\/p>\n<p>&nbsp;<\/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]\\varepsilon >0[\/latex], there exists [latex]N<0[\/latex] such that<\/p>\n<div id=\"fs-id1165042331766\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|f(x)-L|<\\varepsilon[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=L[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div>\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\">Figure 9. For a function with a limit at infinity, for all [latex]x&gt;N[\/latex], [latex]|f(x)-L|&lt;\\varepsilon [\/latex].<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043396243\">Earlier in this section, we used graphical evidence and numerical evidence to conclude that [latex]\\underset{x\\to \\infty }{\\lim}\\left(2+\\frac{1}{x}\\right)=2[\/latex]. Here we use the formal definition of limit at infinity to prove this result rigorously.<\/p>\n<div class=\"textbook exercises\">\n<h3>Example: A Finite Limit at Infinity Example<\/h3>\n<p id=\"fs-id1165042587296\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}\\left(2+\\frac{1}{x}\\right)=2[\/latex].<\/p>\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]\\varepsilon >0[\/latex]. Let [latex]N=\\frac{1}{\\varepsilon }[\/latex]. Therefore, for all [latex]x>N[\/latex], we have<\/p>\n<div id=\"fs-id1165043312498\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left| 2+\\dfrac{1}{x}-2 \\right| =\\left| \\dfrac{1}{x} \\right|=\\dfrac{1}{x}<\\dfrac{1}{N}=\\varepsilon[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: A Finite Limit at Infinity Example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=450&amp;end=630&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.6LimitsAtInfinityAndAsymptotes450to630_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.6 Limits at Infinity and Asymptotes&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1165042480092\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165042480099\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}\\left(3 - \\dfrac{1}{x^2}\\right)=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q9822541\">Hint<\/span><\/p>\n<div id=\"q9822541\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042332065\">Let [latex]N=\\dfrac{1}{\\sqrt{\\varepsilon }}[\/latex].<\/p>\n<\/div>\n<\/div>\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]\\varepsilon >0[\/latex]. Let [latex]N=\\dfrac{1}{\\sqrt{\\varepsilon }}[\/latex]. Therefore, for all [latex]x>N[\/latex], we have<\/p>\n<p id=\"fs-id1165042376362\">[latex]|3-\\dfrac{1}{x^2}-3|=\\dfrac{1}{x^2}<\\dfrac{1}{N^2}=\\varepsilon[\/latex]<\/p>\n<p id=\"fs-id1165042320298\">Therefore, [latex]\\underset{x\\to \\infty }{\\lim}\\left(3-\\frac{1}{x^2}\\right)=3[\/latex].<\/p>\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 shaded\">\n<div class=\"title\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=\\infty[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]f(x)>M[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042374733\">for all [latex]x>N[\/latex] (see Figure 10).<\/p>\n<p>&nbsp;<\/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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=\u2212\\infty[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]f(x)<M[\/latex]<\/div>\n<p>&nbsp;<\/p>\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 \u2212\\infty[\/latex].<\/p>\n<\/div>\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\">Figure 10. 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<p id=\"fs-id1165042705963\">Earlier, we used graphical evidence (Figure 8) and numerical evidence (the table beneath it) to conclude that [latex]\\underset{x\\to \\infty }{\\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=\"textbook exercises\">\n<h3>Example: An Infinite Limit at Infinity<\/h3>\n<p id=\"fs-id1165043395589\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}x^3=\\infty[\/latex].<\/p>\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\" style=\"text-align: center;\">[latex]x^3>N^3=(\\sqrt[3]{M})^3=M[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042604681\">Therefore, [latex]\\underset{x\\to \\infty }{\\lim}x^3=\\infty[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: An Infinite Limit at Infinity.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=666&amp;end=816&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266835\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266835\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.6LimitsAtInfinityAndAsymptotes666to816_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.6 Limits at Infinity and Asymptotes&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1165042323710\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165042320226\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}3x^2=\\infty[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80775166\">Hint<\/span><\/p>\n<div id=\"q80775166\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043219104\">Let [latex]N=\\sqrt{\\frac{M}{3}}[\/latex].<\/p>\n<\/div>\n<\/div>\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]3x^2>3N^2=3(\\sqrt{\\frac{M}{3}})^2=\\frac{3M}{3}=M[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm16109\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=16109&theme=oea&iframe_resize_id=ohm16109&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-407\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>4.6 Limits at Infinity and Asymptotes. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"4.6 Limits at Infinity and Asymptotes\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-407","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/407","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":30,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/407\/revisions"}],"predecessor-version":[{"id":4946,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/407\/revisions\/4946"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/407\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=407"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=407"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=407"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}