{"id":408,"date":"2021-02-04T02:01:58","date_gmt":"2021-02-04T02:01:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=408"},"modified":"2022-03-16T05:47:35","modified_gmt":"2022-03-16T05:47:35","slug":"end-behavior","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/end-behavior\/","title":{"raw":"End Behavior","rendered":"End Behavior"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Estimate the end behavior of a function as \ud835\udc65 increases or decreases without bound<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li>Recognize an oblique asymptote on the graph of a function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165042368492\">The behavior of a function as [latex]x\\to \\pm \\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 \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]\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<h3>End Behavior for Polynomial Functions<\/h3>\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 Figure 11\u00a0and Figure 12, we see that<\/p>\r\n\r\n<div id=\"fs-id1165042545843\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}x^n=\\infty; \\, n=1,2,3, \\cdots[\/latex]<\/div>\r\nand\r\n<div id=\"fs-id1165042705928\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty }{\\lim}x^n=\\begin{cases} \\infty; &amp; n=2,4,6,\\cdots \\\\ -\\infty; &amp; n=1,3,5,\\cdots \\end{cases}[\/latex]<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div>[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\" \/> Figure 11. For power functions with an even power of [latex]n[\/latex], [latex]\\underset{x\\to \\infty }{\\lim}x^n=\\infty =\\underset{x\\to \u2212\\infty }{\\lim}x^n[\/latex].[\/caption]\u00a0[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\" \/> Figure 12. For power functions with an odd power of [latex]n[\/latex], [latex]\\underset{x\\to \\infty }{\\lim}x^n=\\infty [\/latex] and [latex]\\underset{x\\to \u2212\\infty}{\\lim}x^n=\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 }{\\lim}cx^n[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}cx^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=cx^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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}cx^n=\\underset{x\\to \\infty }{\\lim}x^n[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}cx^n=\\underset{x\\to \u2212\\infty}{\\lim}x^n[\/latex] if [latex]c&gt;0[\/latex]<\/div>\r\n<div><\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043424818\">If [latex]c&lt;0[\/latex], the graph of [latex]y=cx^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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}cx^n=\u2212\\underset{x\\to \\infty }{\\lim}x^n[\/latex] and [latex]\\underset{x\\to \u2212\\infty}{\\lim}cx^n=\u2212\\underset{x\\to \u2212\\infty }{\\lim}x^n[\/latex] if [latex]c&lt;0[\/latex]<\/div>\r\n<div><\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042640745\">If [latex]c=0, \\, y=cx^n=0[\/latex], in which case [latex]\\underset{x\\to \\infty }{\\lim}cx^n=0=\\underset{x\\to \u2212\\infty }{\\lim}cx^n[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165043219126\" class=\"textbox exercises\">\r\n<h3>Example: Limits at Infinity for Power Functions<\/h3>\r\nFor each function [latex]f[\/latex], evaluate [latex]\\underset{x\\to \\infty }{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)[\/latex].\r\n<ol id=\"fs-id1165042333246\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(x)=-5x^3[\/latex]<\/li>\r\n \t<li>[latex]f(x)=2x^4[\/latex]<\/li>\r\n<\/ol>\r\n[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)=-5x^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 }{\\lim}(-5x^3)=\u2212\\infty [\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}(-5x^3)=\\infty[\/latex].<\/li>\r\n \t<li>Since the coefficient of [latex]x^4[\/latex] is 2, the graph of [latex]f(x)=2x^4[\/latex] is a vertical stretch of the graph of [latex]y=x^4[\/latex]. Therefore, [latex]\\underset{x\\to \\infty }{\\lim}2x^4=\\infty [\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}2x^4=\\infty[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<div id=\"fs-id1165043219128\" class=\"exercise\"><\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Limits at Infinity for Power Functions.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=877&amp;end=983&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\/4.6LimitsAtInfinityAndAsymptotes877to983_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-id1165042401057\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165042401064\">Let [latex]f(x)=-3x^4[\/latex]. Find [latex]\\underset{x\\to \\infty }{\\lim}f(x)[\/latex].<\/p>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"2288937\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"2288937\"]\r\n\r\n&nbsp;\r\n<p id=\"fs-id1165042708228\">The coefficient -3 is negative.<\/p>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"fs-id1165042708212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042708212\"]\r\n\r\n&nbsp;\r\n<p id=\"fs-id1165042708212\">[latex]\u2212\\infty [\/latex]<\/p>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\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 \\pm \\infty }{\\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\" style=\"text-align: center;\">[latex]f(x)=a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0[\/latex]<\/div>\r\n<div><\/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\" style=\"text-align: center;\">[latex]f(x)=a_n x^n (1+\\frac{a_{n-1}}{a_n}\\frac{1}{x}+ \\cdots + \\frac{a_1}{a_n}\\frac{1}{x^{n-1}} + \\frac{a_0}{a_n}\\frac{1}{x^n})[\/latex].<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165043348532\">As [latex]x\\to \\pm \\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}f(x)=\\underset{x\\to \\pm \\infty }{\\lim} a_n x^n[\/latex].<\/div>\r\n<p id=\"fs-id1165043317360\">For example, the function [latex]f(x)=5x^3-3x^2+4[\/latex] behaves like [latex]g(x)=5x^3[\/latex] as [latex]x\\to \\pm \\infty [\/latex] as shown in Figure 13\u00a0and Table 1.<\/p>\r\n\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\" \/> Figure 13. The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.[\/caption]\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>Table 1. 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)=5x^3-3x^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)=5x^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)=5x^3-3x^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)=5x^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 class=\"bc-section section\">\r\n<h3>End Behavior for Algebraic Functions<\/h3>\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 the example below, 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\nNote that this is not your first encounter with horizontal asymptotes. It may be helpful to recall what you already know about them.\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Horizontal Asymptotes of Rational Functions<\/h3>\r\nThe <strong>horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.\r\n<ul id=\"fs-id1165137722720\">\r\n \t<li><strong>Case 1:<\/strong> Degree of numerator <em>is less than<\/em> degree of denominator: horizontal asymptote at\u00a0[latex]y=0[\/latex]<\/li>\r\n \t<li><strong>Case 2<\/strong>: Degree of numerator <em>is greater than degree of denominator by one<\/em>: no horizontal asymptote; slant asymptote.\r\n<ul>\r\n \t<li>If the degree of the numerator is greater than the degree of the denominator by\u00a0<em>more than one<\/em>, the end behavior of the function's graph will mimic that of the graph of the reduced ratio of leading terms.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Case 3<\/strong>: Degree of numerator <em>is equal to<\/em> degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165042638553\" class=\"textbox exercises\">\r\n<h3>Example: Determining End Behavior for Rational Functions<\/h3>\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 \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{3x^2+2x}{4x^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{3x^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[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\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\underset{x\\to \\pm \\infty }{\\lim}\\frac{3x-1}{2x+5} &amp; =\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3-1\/x}{2+5\/x} \\\\ &amp; =\\frac{\\underset{x\\to \\pm \\infty }{\\lim}(3-1\/x)}{\\underset{x\\to \\pm \\infty }{\\lim}(2+5\/x)} \\\\ &amp; =\\frac{\\underset{x\\to \\pm \\infty }{\\lim}3-\\underset{x\\to \\pm \\infty }{\\lim}1\/x}{\\underset{x\\to \\pm \\infty }{\\lim}2+\\underset{x\\to \\pm \\infty }{\\lim}5\/x} \\\\ &amp; =\\frac{3-0}{2+0}=\\frac{3}{2}. \\end{array}[\/latex]<\/div>\r\nSince [latex]\\underset{x\\to \\pm \\infty }{\\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\r\n[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\" \/> Figure 14. The graph of this rational function approaches a horizontal asymptote as [latex]x\\to \\pm \\infty[\/latex].[\/caption]<\/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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3x^2+2x}{4x^3-5x+7}=\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3\/x+2\/x^2}{4-5\/x^2+7\/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\r\n[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\" \/> Figure 15. The graph of this rational function approaches the horizontal asymptote [latex]y=0[\/latex] as [latex]x\\to \\pm \\infty[\/latex].[\/caption]<\/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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3x^2+4x}{x+2}=\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3x+4}{1+2\/x}[\/latex].<\/div>\r\nAs [latex]x\\to \\pm \\infty[\/latex], the denominator approaches 1. As [latex]x\\to \\infty[\/latex], the numerator approaches [latex]+\\infty[\/latex]. As [latex]x\\to \u2212\\infty[\/latex], the numerator approaches [latex]\u2212\\infty[\/latex]. Therefore [latex]\\underset{x\\to \\infty }{\\lim}f(x)=\\infty[\/latex], whereas [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=\u2212\\infty [\/latex] as shown in the following figure.\r\n\r\n[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\" \/> Figure 16. As [latex]x\\to \\infty[\/latex], the values [latex]f(x)\\to \\infty[\/latex]. As [latex]x\\to \u2212\\infty[\/latex], the values [latex]f(x)\\to \u2212\\infty[\/latex].[\/caption]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<div id=\"fs-id1165042638555\" class=\"exercise\"><\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Determining End Behavior for Rational Functions.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=984&amp;end=1278&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\/4.6LimitsAtInfinityAndAsymptotes984to1278_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-id1165042660288\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165042660296\">Evaluate [latex]\\underset{x\\to \\pm \\infty }{\\lim}\\dfrac{3x^2+2x-1}{5x^2-4x+7}[\/latex] and use these limits to determine the end behavior of [latex]f(x)=\\dfrac{3x^2+2x-1}{5x^2-4x+7}[\/latex].<\/p>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"4338920\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"4338920\"]\r\n\r\n&nbsp;\r\n<p id=\"fs-id1165042374889\">Divide the numerator and denominator by [latex]x^2[\/latex].<\/p>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"fs-id1165042374871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042374871\"]\r\n\r\n&nbsp;\r\n<p id=\"fs-id1165042374871\">[latex]\\frac{3}{5}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042374905\">Before proceeding, consider the graph of [latex]f(x)=\\frac{(3x^2+4x)}{(x+2)}[\/latex] shown in Figure 17. As [latex]x\\to \\infty [\/latex] and [latex]x\\to \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\" style=\"text-align: center;\">[latex]f(x)=\\frac{3x^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 \\pm \\infty[\/latex], we conclude that<\/p>\r\n\r\n<div id=\"fs-id1165042465555\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}(f(x)-(3x-2))=\\underset{x\\to \\pm \\infty }{\\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 \\pm \\infty[\/latex]. This line is known as an <strong>oblique asymptote<\/strong> for [latex]f[\/latex] (Figure 17).<\/p>\r\n\r\n[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)=(3x^2+4x)\/(x+2)[\/latex] approaches the oblique asymptote [latex]y=3x-2[\/latex] as [latex]x\\to \\pm \\infty[\/latex].[\/caption]\r\n<p id=\"fs-id1165042461217\">We can summarize the results of the example above\u00a0to 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\" style=\"text-align: center;\">[latex]f(x)=\\frac{p(x)}{q(x)}=\\frac{a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \\cdots + b_1 x + b_0}[\/latex],<\/div>\r\n<p id=\"fs-id1165042315666\">where [latex]a_n\\ne 0[\/latex] and [latex]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\/b_m[\/latex] as [latex]x\\to \\pm \\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 \\pm \\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 \\pm \\infty }{\\lim}r(x)\/q(x)=0[\/latex]. Therefore, the values of [latex][f(x)-g(x)][\/latex] approach zero as [latex]x\\to \\pm \\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 exercises\">\r\n<h3>Example: Determining End Behavior for a Function Involving a Radical<\/h3>\r\nFind the limits as [latex]x\\to \\infty [\/latex] and [latex]x\\to \u2212\\infty [\/latex] for [latex]f(x)=\\frac{3x-2}{\\sqrt{4x^2+5}}[\/latex] and describe the end behavior of [latex]f[\/latex].\r\n\r\n[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{4x^2+5}[\/latex] in the denominator. Since<\/p>\r\n\r\n<div id=\"fs-id1165042418061\" class=\"equation unnumbered\">[latex]\\sqrt{4x^2+5}\\approx \\sqrt{4x^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|=\u2212x[\/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\" style=\"text-align: center;\">[latex]\\begin{array}{lll} \\underset{x\\to \\infty }{\\lim}\\frac{3x-2}{\\sqrt{4x^2+5}} &amp; = &amp; \\underset{x\\to \\infty }{\\lim}\\frac{(1\/|x|)(3x-2)}{(1\/|x|)\\sqrt{4x^2+5}} \\\\ &amp; = &amp; \\underset{x\\to \\infty }{\\lim}\\frac{(1\/x)(3x-2)}{\\sqrt{(1\/x^2)(4x^2+5)}} \\\\ &amp; = &amp; \\underset{x\\to \\infty }{\\lim}\\frac{3-2\/x}{\\sqrt{4+5\/x^2}}=\\frac{3}{\\sqrt{4}}=\\frac{3}{2} \\\\ \\underset{x\\to \u2212\\infty }{\\lim}\\frac{3x-2}{\\sqrt{4x^2+5}} &amp; = &amp; \\underset{x\\to \u2212\\infty }{\\lim}\\frac{(1\/|x|)(3x-2)}{(1\/|x|)\\sqrt{4x^2+5}} \\\\ &amp; = &amp; \\underset{x\\to \u2212\\infty }{\\lim}\\frac{(-1\/x)(3x-2)}{\\sqrt{(1\/x^2)(4x^2+5)}} \\\\ &amp; = &amp; \\underset{x\\to \u2212\\infty }{\\lim}\\frac{-3+2\/x}{\\sqrt{4+5\/x^2}}=\\frac{-3}{\\sqrt{4}}=\\frac{-3}{2}. \\end{array}[\/latex]<\/div>\r\n&nbsp;\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 \u2212\\infty [\/latex] as shown in the following graph.<\/p>\r\n&nbsp;\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\" \/> Figure 18. This function has two horizontal asymptotes and it crosses one of the asymptotes.[\/caption]\r\n\r\n[\/hidden-answer]\r\n<div id=\"fs-id1165042631818\" class=\"exercise\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042463914\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165042463923\">Evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\sqrt{3x^2+4}}{x+6}[\/latex].<\/p>\r\n[reveal-answer q=\"1990672\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"1990672\"]\r\n<p id=\"fs-id1165043327311\">Divide the numerator and denominator by [latex]|x|[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165043327293\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043327293\"]\r\n<p id=\"fs-id1165043327293\">[latex]\\pm \\sqrt{3}[\/latex]<\/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]57920[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043327330\" class=\"bc-section section\">\r\n<h3>Determining End Behavior for Transcendental Functions<\/h3>\r\n<p id=\"fs-id1165043327335\">The six basic trigonometric functions are periodic and do not approach a finite limit as [latex]x\\to \\pm \\infty[\/latex]. For example, [latex]\\sin x[\/latex] oscillates between 1 and -1 (Figure 19). The tangent function [latex]x[\/latex] has an infinite number of vertical asymptotes as [latex]x\\to \\pm \\infty[\/latex]; therefore, it does not approach a finite limit nor does it approach [latex]\\pm \\infty [\/latex] as [latex]x\\to \\pm \\infty [\/latex] as shown in (Figure 20).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043327330\" class=\"bc-section section\">[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\" \/> Figure 19. The function [latex]f(x)= \\sin x[\/latex] oscillates between 1 and -1 as [latex]x\\to \\pm \\infty [\/latex][\/caption]\u00a0[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\" \/> Figure 20. The function [latex]f(x)= \\tan x[\/latex] does not approach a limit and does not approach [latex]\\pm \\infty [\/latex] as [latex]x\\to \\pm \\infty [\/latex][\/caption]\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](\u2212\\infty ,\\infty )[\/latex] and range [latex](0,\\infty )[\/latex]. If [latex]b&gt;1, \\, y=b^x[\/latex] is increasing over [latex](\u2212\\infty ,\\infty )[\/latex]. If [latex]0&lt;b&lt;1[\/latex], [latex]y=b^x[\/latex] is decreasing over [latex](\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](\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 \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><\/div>\r\n[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\" \/> Figure 21. The exponential function approaches zero as [latex]x\\to \u2212\\infty [\/latex] and approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty[\/latex].[\/caption]\r\n<p id=\"fs-id1165043218124\">Recall that the natural logarithm function [latex]f(x)=\\ln (x)[\/latex] is the inverse of the natural exponential function [latex]y=e^x[\/latex]. Therefore, the domain of [latex]f(x)=\\ln (x)[\/latex] is [latex](0,\\infty )[\/latex] and the range is [latex](\u2212\\infty ,\\infty )[\/latex]. The graph of [latex]f(x)=\\ln (x)[\/latex] is the reflection of the graph of [latex]y=e^x[\/latex] about the line [latex]y=x[\/latex]. Therefore, [latex]\\ln (x)\\to \u2212\\infty [\/latex] as [latex]x\\to 0^+[\/latex] and [latex]\\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]\\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><\/div>\r\n[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\" \/> Figure 22. The natural logarithm function approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty[\/latex].[\/caption]\r\n<div id=\"fs-id1165042469818\" class=\"textbox exercises\">\r\n<h3>example: Determining End Behavior for a Transcendental Function<\/h3>\r\nFind the limits as [latex]x\\to \\infty [\/latex] and [latex]x\\to \u2212\\infty [\/latex] for [latex]f(x)=\\frac{(2+3e^x)}{(7-5e^x)}[\/latex] and describe the end behavior of [latex]f[\/latex].\r\n\r\n[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[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1165042711652\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\underset{x\\to \\infty }{\\lim}f(x) &amp; =\\underset{x\\to \\infty }{\\lim}\\frac{2+3e^x}{7-5e^x} \\\\ &amp; =\\underset{x\\to \\infty }{\\lim}\\frac{(2\/e^x)+3}{(7\/e^x)-5}. \\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165042499478\">As shown in Figure 21, [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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{2}{e^x}=0=\\underset{x\\to \\infty }{\\lim}\\frac{7}{e^x}[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042499577\">We conclude that [latex]\\underset{x\\to \\infty }{\\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 \u2212\\infty[\/latex], use the fact that [latex]e^x \\to 0[\/latex] as [latex]x\\to \u2212\\infty [\/latex] to conclude that [latex]\\underset{x\\to -\\infty }{\\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 \u2212\\infty[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Determining End Behavior for a Transcendental Function.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=1453&amp;end=1547&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\/4.6LimitsAtInfinityAndAsymptotes1453to1547_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-id1165042711306\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165042711314\">Find the limits as [latex]x\\to \\infty [\/latex] and [latex]x\\to \u2212\\infty [\/latex] for [latex]f(x)=\\dfrac{(3e^x-4)}{(5e^x+2)}[\/latex].<\/p>\r\n[reveal-answer q=\"377625\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"377625\"]\r\n<p id=\"fs-id1165042711480\">[latex]\\underset{x\\to \\infty }{\\lim}e^x=\\infty [\/latex] and [latex]\\underset{x\\to -\\infty }{\\lim}e^x=0[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[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 }{\\lim}f(x)=\\frac{3}{5}[\/latex], [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=-2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042602932\" class=\"bc-section section\"><\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Estimate the end behavior of a function as \ud835\udc65 increases or decreases without bound<\/li>\n<\/ul>\n<ul>\n<li>Recognize an oblique asymptote on the graph of a function<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165042368492\">The behavior of a function as [latex]x\\to \\pm \\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 \u2212\\infty[\/latex].<\/li>\n<li>The function does not approach a finite limit, nor does it approach [latex]\\infty[\/latex] or [latex]\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<h3>End Behavior for Polynomial Functions<\/h3>\n<p id=\"fs-id1165042323672\">Consider the power function [latex]f(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer. From Figure 11\u00a0and Figure 12, we see that<\/p>\n<div id=\"fs-id1165042545843\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}x^n=\\infty; \\, n=1,2,3, \\cdots[\/latex]<\/div>\n<p>and<\/p>\n<div id=\"fs-id1165042705928\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty }{\\lim}x^n=\\begin{cases} \\infty; & n=2,4,6,\\cdots \\\\ -\\infty; & n=1,3,5,\\cdots \\end{cases}[\/latex]<\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div>\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\">Figure 11. For power functions with an even power of [latex]n[\/latex], [latex]\\underset{x\\to \\infty }{\\lim}x^n=\\infty =\\underset{x\\to \u2212\\infty }{\\lim}x^n[\/latex].<\/p>\n<\/div>\n<p>\u00a0<\/p>\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\">Figure 12. For power functions with an odd power of [latex]n[\/latex], [latex]\\underset{x\\to \\infty }{\\lim}x^n=\\infty [\/latex] and [latex]\\underset{x\\to \u2212\\infty}{\\lim}x^n=\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 }{\\lim}cx^n[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}cx^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=cx^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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}cx^n=\\underset{x\\to \\infty }{\\lim}x^n[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}cx^n=\\underset{x\\to \u2212\\infty}{\\lim}x^n[\/latex] if [latex]c>0[\/latex]<\/div>\n<div><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043424818\">If [latex]c<0[\/latex], the graph of [latex]y=cx^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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}cx^n=\u2212\\underset{x\\to \\infty }{\\lim}x^n[\/latex] and [latex]\\underset{x\\to \u2212\\infty}{\\lim}cx^n=\u2212\\underset{x\\to \u2212\\infty }{\\lim}x^n[\/latex] if [latex]c<0[\/latex]<\/div>\n<div><\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042640745\">If [latex]c=0, \\, y=cx^n=0[\/latex], in which case [latex]\\underset{x\\to \\infty }{\\lim}cx^n=0=\\underset{x\\to \u2212\\infty }{\\lim}cx^n[\/latex].<\/p>\n<div id=\"fs-id1165043219126\" class=\"textbox exercises\">\n<h3>Example: Limits at Infinity for Power Functions<\/h3>\n<p>For each function [latex]f[\/latex], evaluate [latex]\\underset{x\\to \\infty }{\\lim}f(x)[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)[\/latex].<\/p>\n<ol id=\"fs-id1165042333246\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(x)=-5x^3[\/latex]<\/li>\n<li>[latex]f(x)=2x^4[\/latex]<\/li>\n<\/ol>\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)=-5x^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 }{\\lim}(-5x^3)=\u2212\\infty[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}(-5x^3)=\\infty[\/latex].<\/li>\n<li>Since the coefficient of [latex]x^4[\/latex] is 2, the graph of [latex]f(x)=2x^4[\/latex] is a vertical stretch of the graph of [latex]y=x^4[\/latex]. Therefore, [latex]\\underset{x\\to \\infty }{\\lim}2x^4=\\infty[\/latex] and [latex]\\underset{x\\to \u2212\\infty }{\\lim}2x^4=\\infty[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043219128\" class=\"exercise\"><\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Limits at Infinity for Power Functions.<\/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=877&amp;end=983&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\/4.6LimitsAtInfinityAndAsymptotes877to983_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-id1165042401057\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165042401064\">Let [latex]f(x)=-3x^4[\/latex]. Find [latex]\\underset{x\\to \\infty }{\\lim}f(x)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q2288937\">Hint<\/span><\/p>\n<div id=\"q2288937\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042708228\">The coefficient -3 is negative.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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>&nbsp;<\/p>\n<p id=\"fs-id1165042708212\">[latex]\u2212\\infty[\/latex]<\/p>\n<p>&nbsp;<\/p>\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 \\pm \\infty }{\\lim}f(x)[\/latex] for any polynomial function [latex]f[\/latex]. Consider a polynomial function<\/p>\n<div id=\"fs-id1165042710943\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)=a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0[\/latex]<\/div>\n<div><\/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\" style=\"text-align: center;\">[latex]f(x)=a_n x^n (1+\\frac{a_{n-1}}{a_n}\\frac{1}{x}+ \\cdots + \\frac{a_1}{a_n}\\frac{1}{x^{n-1}} + \\frac{a_0}{a_n}\\frac{1}{x^n})[\/latex].<\/div>\n<div><\/div>\n<p id=\"fs-id1165043348532\">As [latex]x\\to \\pm \\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}f(x)=\\underset{x\\to \\pm \\infty }{\\lim} a_n x^n[\/latex].<\/div>\n<p id=\"fs-id1165043317360\">For example, the function [latex]f(x)=5x^3-3x^2+4[\/latex] behaves like [latex]g(x)=5x^3[\/latex] as [latex]x\\to \\pm \\infty[\/latex] as shown in Figure 13\u00a0and Table 1.<\/p>\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\">Figure 13. The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.<\/p>\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>Table 1. 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)=5x^3-3x^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)=5x^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)=5x^3-3x^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)=5x^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 class=\"bc-section section\">\n<h3>End Behavior for Algebraic Functions<\/h3>\n<p id=\"fs-id1165042638493\">The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In the example below, 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<p>Note that this is not your first encounter with horizontal asymptotes. It may be helpful to recall what you already know about them.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Horizontal Asymptotes of Rational Functions<\/h3>\n<p>The <strong>horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.<\/p>\n<ul id=\"fs-id1165137722720\">\n<li><strong>Case 1:<\/strong> Degree of numerator <em>is less than<\/em> degree of denominator: horizontal asymptote at\u00a0[latex]y=0[\/latex]<\/li>\n<li><strong>Case 2<\/strong>: Degree of numerator <em>is greater than degree of denominator by one<\/em>: no horizontal asymptote; slant asymptote.\n<ul>\n<li>If the degree of the numerator is greater than the degree of the denominator by\u00a0<em>more than one<\/em>, the end behavior of the function&#8217;s graph will mimic that of the graph of the reduced ratio of leading terms.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Case 3<\/strong>: Degree of numerator <em>is equal to<\/em> degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165042638553\" class=\"textbox exercises\">\n<h3>Example: Determining End Behavior for Rational Functions<\/h3>\n<p id=\"fs-id1165042638562\">For each of the following functions, determine the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \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{3x^2+2x}{4x^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{3x^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 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\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\underset{x\\to \\pm \\infty }{\\lim}\\frac{3x-1}{2x+5} & =\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3-1\/x}{2+5\/x} \\\\ & =\\frac{\\underset{x\\to \\pm \\infty }{\\lim}(3-1\/x)}{\\underset{x\\to \\pm \\infty }{\\lim}(2+5\/x)} \\\\ & =\\frac{\\underset{x\\to \\pm \\infty }{\\lim}3-\\underset{x\\to \\pm \\infty }{\\lim}1\/x}{\\underset{x\\to \\pm \\infty }{\\lim}2+\\underset{x\\to \\pm \\infty }{\\lim}5\/x} \\\\ & =\\frac{3-0}{2+0}=\\frac{3}{2}. \\end{array}[\/latex]<\/div>\n<p>Since [latex]\\underset{x\\to \\pm \\infty }{\\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 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\">Figure 14. The graph of this rational function approaches a horizontal asymptote as [latex]x\\to \\pm \\infty[\/latex].<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3x^2+2x}{4x^3-5x+7}=\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3\/x+2\/x^2}{4-5\/x^2+7\/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 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\">Figure 15. The graph of this rational function approaches the horizontal asymptote [latex]y=0[\/latex] as [latex]x\\to \\pm \\infty[\/latex].<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3x^2+4x}{x+2}=\\underset{x\\to \\pm \\infty }{\\lim}\\frac{3x+4}{1+2\/x}[\/latex].<\/div>\n<p>As [latex]x\\to \\pm \\infty[\/latex], the denominator approaches 1. As [latex]x\\to \\infty[\/latex], the numerator approaches [latex]+\\infty[\/latex]. As [latex]x\\to \u2212\\infty[\/latex], the numerator approaches [latex]\u2212\\infty[\/latex]. Therefore [latex]\\underset{x\\to \\infty }{\\lim}f(x)=\\infty[\/latex], whereas [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=\u2212\\infty[\/latex] as shown in the following figure.<\/p>\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\">Figure 16. As [latex]x\\to \\infty[\/latex], the values [latex]f(x)\\to \\infty[\/latex]. As [latex]x\\to \u2212\\infty[\/latex], the values [latex]f(x)\\to \u2212\\infty[\/latex].<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042638555\" class=\"exercise\"><\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Determining End Behavior for Rational Functions.<\/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=984&amp;end=1278&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\/4.6LimitsAtInfinityAndAsymptotes984to1278_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-id1165042660288\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165042660296\">Evaluate [latex]\\underset{x\\to \\pm \\infty }{\\lim}\\dfrac{3x^2+2x-1}{5x^2-4x+7}[\/latex] and use these limits to determine the end behavior of [latex]f(x)=\\dfrac{3x^2+2x-1}{5x^2-4x+7}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4338920\">Hint<\/span><\/p>\n<div id=\"q4338920\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042374889\">Divide the numerator and denominator by [latex]x^2[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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>&nbsp;<\/p>\n<p id=\"fs-id1165042374871\">[latex]\\frac{3}{5}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042374905\">Before proceeding, consider the graph of [latex]f(x)=\\frac{(3x^2+4x)}{(x+2)}[\/latex] shown in Figure 17. As [latex]x\\to \\infty[\/latex] and [latex]x\\to \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\" style=\"text-align: center;\">[latex]f(x)=\\frac{3x^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 \\pm \\infty[\/latex], we conclude that<\/p>\n<div id=\"fs-id1165042465555\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}(f(x)-(3x-2))=\\underset{x\\to \\pm \\infty }{\\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 \\pm \\infty[\/latex]. This line is known as an <strong>oblique asymptote<\/strong> for [latex]f[\/latex] (Figure 17).<\/p>\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)=(3x^2+4x)\/(x+2)[\/latex] approaches the oblique asymptote [latex]y=3x-2[\/latex] as [latex]x\\to \\pm \\infty[\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1165042461217\">We can summarize the results of the example above\u00a0to make the following conclusion regarding end behavior for rational functions. Consider a rational function<\/p>\n<div id=\"fs-id1165042461226\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)=\\frac{p(x)}{q(x)}=\\frac{a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \\cdots + b_1 x + b_0}[\/latex],<\/div>\n<p id=\"fs-id1165042315666\">where [latex]a_n\\ne 0[\/latex] and [latex]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\/b_m[\/latex] as [latex]x\\to \\pm \\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 \\pm \\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 \\pm \\infty }{\\lim}r(x)\/q(x)=0[\/latex]. Therefore, the values of [latex][f(x)-g(x)][\/latex] approach zero as [latex]x\\to \\pm \\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 exercises\">\n<h3>Example: Determining End Behavior for a Function Involving a Radical<\/h3>\n<p>Find the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \u2212\\infty[\/latex] for [latex]f(x)=\\frac{3x-2}{\\sqrt{4x^2+5}}[\/latex] and describe the end behavior of [latex]f[\/latex].<\/p>\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{4x^2+5}[\/latex] in the denominator. Since<\/p>\n<div id=\"fs-id1165042418061\" class=\"equation unnumbered\">[latex]\\sqrt{4x^2+5}\\approx \\sqrt{4x^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|=\u2212x[\/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\" style=\"text-align: center;\">[latex]\\begin{array}{lll} \\underset{x\\to \\infty }{\\lim}\\frac{3x-2}{\\sqrt{4x^2+5}} & = & \\underset{x\\to \\infty }{\\lim}\\frac{(1\/|x|)(3x-2)}{(1\/|x|)\\sqrt{4x^2+5}} \\\\ & = & \\underset{x\\to \\infty }{\\lim}\\frac{(1\/x)(3x-2)}{\\sqrt{(1\/x^2)(4x^2+5)}} \\\\ & = & \\underset{x\\to \\infty }{\\lim}\\frac{3-2\/x}{\\sqrt{4+5\/x^2}}=\\frac{3}{\\sqrt{4}}=\\frac{3}{2} \\\\ \\underset{x\\to \u2212\\infty }{\\lim}\\frac{3x-2}{\\sqrt{4x^2+5}} & = & \\underset{x\\to \u2212\\infty }{\\lim}\\frac{(1\/|x|)(3x-2)}{(1\/|x|)\\sqrt{4x^2+5}} \\\\ & = & \\underset{x\\to \u2212\\infty }{\\lim}\\frac{(-1\/x)(3x-2)}{\\sqrt{(1\/x^2)(4x^2+5)}} \\\\ & = & \\underset{x\\to \u2212\\infty }{\\lim}\\frac{-3+2\/x}{\\sqrt{4+5\/x^2}}=\\frac{-3}{\\sqrt{4}}=\\frac{-3}{2}. \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\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 \u2212\\infty[\/latex] as shown in the following graph.<\/p>\n<p>&nbsp;<\/p>\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\">Figure 18. This function has two horizontal asymptotes and it crosses one of the asymptotes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042631818\" class=\"exercise\"><\/div>\n<\/div>\n<div id=\"fs-id1165042463914\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165042463923\">Evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\sqrt{3x^2+4}}{x+6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q1990672\">Hint<\/span><\/p>\n<div id=\"q1990672\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043327311\">Divide the numerator and denominator by [latex]|x|[\/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-id1165043327293\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043327293\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043327293\">[latex]\\pm \\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm57920\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=57920&theme=oea&iframe_resize_id=ohm57920&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165043327330\" class=\"bc-section section\">\n<h3>Determining End Behavior for Transcendental Functions<\/h3>\n<p id=\"fs-id1165043327335\">The six basic trigonometric functions are periodic and do not approach a finite limit as [latex]x\\to \\pm \\infty[\/latex]. For example, [latex]\\sin x[\/latex] oscillates between 1 and -1 (Figure 19). The tangent function [latex]x[\/latex] has an infinite number of vertical asymptotes as [latex]x\\to \\pm \\infty[\/latex]; therefore, it does not approach a finite limit nor does it approach [latex]\\pm \\infty[\/latex] as [latex]x\\to \\pm \\infty[\/latex] as shown in (Figure 20).<\/p>\n<\/div>\n<div id=\"fs-id1165043327330\" class=\"bc-section section\">\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\">Figure 19. The function [latex]f(x)= \\sin x[\/latex] oscillates between 1 and -1 as [latex]x\\to \\pm \\infty [\/latex]<\/p>\n<\/div>\n<p>\u00a0<\/p>\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\">Figure 20. The function [latex]f(x)= \\tan x[\/latex] does not approach a limit and does not approach [latex]\\pm \\infty [\/latex] as [latex]x\\to \\pm \\infty [\/latex]<\/p>\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](\u2212\\infty ,\\infty )[\/latex] and range [latex](0,\\infty )[\/latex]. If [latex]b>1, \\, y=b^x[\/latex] is increasing over [latex](\u2212\\infty ,\\infty )[\/latex]. If [latex]0<b<1[\/latex], [latex]y=b^x[\/latex] is decreasing over [latex](\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](\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 \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><\/div>\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\">Figure 21. The exponential function approaches zero as [latex]x\\to \u2212\\infty [\/latex] and approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty[\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1165043218124\">Recall that the natural logarithm function [latex]f(x)=\\ln (x)[\/latex] is the inverse of the natural exponential function [latex]y=e^x[\/latex]. Therefore, the domain of [latex]f(x)=\\ln (x)[\/latex] is [latex](0,\\infty )[\/latex] and the range is [latex](\u2212\\infty ,\\infty )[\/latex]. The graph of [latex]f(x)=\\ln (x)[\/latex] is the reflection of the graph of [latex]y=e^x[\/latex] about the line [latex]y=x[\/latex]. Therefore, [latex]\\ln (x)\\to \u2212\\infty[\/latex] as [latex]x\\to 0^+[\/latex] and [latex]\\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]\\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><\/div>\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\">Figure 22. The natural logarithm function approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165042469818\" class=\"textbox exercises\">\n<h3>example: Determining End Behavior for a Transcendental Function<\/h3>\n<p>Find the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \u2212\\infty[\/latex] for [latex]f(x)=\\frac{(2+3e^x)}{(7-5e^x)}[\/latex] and describe the end behavior of [latex]f[\/latex].<\/p>\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[\/latex]:<\/p>\n<div id=\"fs-id1165042711652\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\underset{x\\to \\infty }{\\lim}f(x) & =\\underset{x\\to \\infty }{\\lim}\\frac{2+3e^x}{7-5e^x} \\\\ & =\\underset{x\\to \\infty }{\\lim}\\frac{(2\/e^x)+3}{(7\/e^x)-5}. \\end{array}[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165042499478\">As shown in Figure 21, [latex]e^x\\to \\infty[\/latex] as [latex]x\\to \\infty[\/latex]. Therefore,<\/p>\n<div id=\"fs-id1165042499514\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{2}{e^x}=0=\\underset{x\\to \\infty }{\\lim}\\frac{7}{e^x}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042499577\">We conclude that [latex]\\underset{x\\to \\infty }{\\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 \u2212\\infty[\/latex], use the fact that [latex]e^x \\to 0[\/latex] as [latex]x\\to \u2212\\infty[\/latex] to conclude that [latex]\\underset{x\\to -\\infty }{\\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 \u2212\\infty[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Determining End Behavior for a Transcendental Function.<\/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=1453&amp;end=1547&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\/4.6LimitsAtInfinityAndAsymptotes1453to1547_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-id1165042711306\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165042711314\">Find the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \u2212\\infty[\/latex] for [latex]f(x)=\\dfrac{(3e^x-4)}{(5e^x+2)}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q377625\">Hint<\/span><\/p>\n<div id=\"q377625\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711480\">[latex]\\underset{x\\to \\infty }{\\lim}e^x=\\infty[\/latex] and [latex]\\underset{x\\to -\\infty }{\\lim}e^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-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 }{\\lim}f(x)=\\frac{3}{5}[\/latex], [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042602932\" class=\"bc-section section\"><\/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-408\">\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":21,"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 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