{"id":312,"date":"2023-09-20T22:49:00","date_gmt":"2023-09-20T22:49:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/limits-at-infinity\/"},"modified":"2024-08-05T13:16:48","modified_gmt":"2024-08-05T13:16:48","slug":"limits-at-infinity-and-asymptotes-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/limits-at-infinity-and-asymptotes-learn-it-1\/","title":{"raw":"Limits at Infinity and Asymptotes: Learn It 1","rendered":"Limits at Infinity and Asymptotes: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Determine limits and predict how functions behave as x increases or decreases indefinitely<\/li>\r\n\t<li>Identify and distinguish horizontal and slanting lines that a graph approaches but never touches<\/li>\r\n\t<li>Use a function\u2019s derivatives to accurately sketch its graph<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Limits at Infinity<\/h2>\r\n<p>We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function [latex]f[\/latex] defined on an unbounded domain, we also need to know the behavior of [latex]f[\/latex] as [latex]x \\to \\pm \\infty[\/latex]. In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function [latex]f[\/latex].<\/p>\r\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>\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.<\/p>\r\n<section class=\"textbox example\">\r\n<p>Consider the function [latex]f(x)=2+\\frac{1}{x}[\/latex].<\/p>\r\n<p>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 [latex]2[\/latex]. We say the limit as [latex]x[\/latex] approaches [latex]\\infty [\/latex] of [latex]f(x)[\/latex] is [latex]2[\/latex] and write:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=2[\/latex].<\/p>\r\n<p>Similarly, for [latex]x&lt;0[\/latex], as the values [latex]|x|[\/latex] get larger, the values of [latex]f(x)[\/latex] approaches [latex]2[\/latex]. We say the limit as [latex]x[\/latex] approaches [latex]\u2212\\infty [\/latex] of [latex]f(x)[\/latex] is [latex]2[\/latex] and write:<\/p>\r\n<p style=\"text-align: center;\">[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\r\n<table id=\"fs-id1165043428402\" class=\"column-header aligncenter\" 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.\">\r\n<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>[latex]10[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<td>[latex]1,000[\/latex]<\/td>\r\n<td>[latex]10,000[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]2+\\frac{1}{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]2.1[\/latex]<\/td>\r\n<td>[latex]2.01[\/latex]<\/td>\r\n<td>[latex]2.001[\/latex]<\/td>\r\n<td>[latex]2.0001[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]-10[\/latex]<\/td>\r\n<td>[latex]-100[\/latex]<\/td>\r\n<td>[latex]-1000[\/latex]<\/td>\r\n<td>[latex]-10,000[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]2+\\frac{1}{x}[\/latex]<\/strong><\/td>\r\n<td>[latex]1.9[\/latex]<\/td>\r\n<td>[latex]1.99[\/latex]<\/td>\r\n<td>[latex]1.999[\/latex]<\/td>\r\n<td>[latex]1.9999[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n<p>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:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to \\infty}{\\lim}f(x)=L[\/latex].<\/p>\r\n<p>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:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=L[\/latex].<\/p>\r\n<p>We now look at the definition of a function having a limit at infinity.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">limit at infinity (informal)<\/h3>\r\n<p id=\"fs-id1165042970725\">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<div id=\"fs-id1165042986551\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\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<div id=\"fs-id1165043105208\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to -\\infty }{\\lim}f(x)=L[\/latex]<\/div>\r\n<\/section>\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].<\/p>\r\n<section class=\"textbox example\">\r\n<p>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<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>horizontal asymptote<\/h3>\r\n<p>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<\/section>\r\n\r\n[caption id=\"\" align=\"alignnone\" 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: 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 -\\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\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.<\/p>\r\n<section class=\"textbox example\">\r\n<p>The function [latex]f(x)=\\frac{ \\cos x}{x}+1[\/latex] shown below 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<\/section>\r\n<p id=\"fs-id1165042373486\">The algebraic limit laws and squeeze theorem we introduced earlier\u00a0also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.<\/p>\r\n<section class=\"textbox example\">\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<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<p>[reveal-answer q=\"fs-id1165043183885\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043183885\"]<\/p>\r\n<ol id=\"fs-id1165043183885\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Using the algebraic limit laws, we have:<center>[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].<\/center>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.\r\n[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]\r\n<\/li>\r\n\t<li>Since [latex]-1\\le \\sin x\\le 1[\/latex] for all [latex]x[\/latex], we have:<br \/>\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\n<p>for all [latex]x \\ne 0[\/latex]. Also, since,<\/p>\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\n<p>we can apply the squeeze theorem to conclude that:<\/p>\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\n<p>Similarly,<\/p>\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<p>&nbsp;<\/p>\r\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>\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]\r\n<\/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.\r\n[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]\r\n<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165043092430\">Since [latex]\\underset{x\\to (\\pi\/2)^-}{\\lim} \\tan x=\\infty [\/latex], it follows that:<\/p>\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<p id=\"fs-id1165043097156\">Similarly, since [latex]\\underset{x\\to (\\pi\/2)^+}{\\lim} \\tan x=\u2212\\infty[\/latex], it follows that:<\/p>\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<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>\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<p>Watch the following video to see the worked solution to this example.<\/p>\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\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>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>\r\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 \"4.6 Limits at Infinity and Asymptotes\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]169165[\/ohm_question]<\/p>\r\n<\/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].<\/p>\r\n<p>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<section class=\"textbox example\">\r\n<p id=\"fs-id1165042606820\">Consider the function [latex]f(x)=x^3[\/latex].<\/p>\r\n\r\n[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]\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.\">\r\n<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>[latex]10[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]50[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<td>[latex]1000[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x^3[\/latex]<\/strong><\/td>\r\n<td>[latex]1000[\/latex]<\/td>\r\n<td>[latex]8000[\/latex]<\/td>\r\n<td>[latex]125,000[\/latex]<\/td>\r\n<td>[latex]1,000,000[\/latex]<\/td>\r\n<td>[latex]1,000,000,000[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]-10[\/latex]<\/td>\r\n<td>[latex]-20[\/latex]<\/td>\r\n<td>[latex]-50[\/latex]<\/td>\r\n<td>[latex]-100[\/latex]<\/td>\r\n<td>[latex]-1000[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x^3[\/latex]<\/strong><\/td>\r\n<td>[latex]-1000[\/latex]<\/td>\r\n<td>[latex]-8000[\/latex]<\/td>\r\n<td>[latex]-125,000[\/latex]<\/td>\r\n<td>[latex]-1,000,000[\/latex]<\/td>\r\n<td>[latex]-1,000,000,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>As seen in the table and figure above, 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].<\/p>\r\n<p>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<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div class=\"title\">\r\n<h3 style=\"text-align: left;\">infinite limits at infinity (informal)<\/h3>\r\n<\/div>\r\n<p id=\"fs-id1165043276356\">We say a function [latex]f[\/latex] has an infinite limit at infinity and write:<\/p>\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<p>&nbsp;<\/p>\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<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<p>&nbsp;<\/p>\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.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Similarly, we can define infinite limits as [latex]x\\to \u2212\\infty[\/latex].<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Determine limits and predict how functions behave as x increases or decreases indefinitely<\/li>\n<li>Identify and distinguish horizontal and slanting lines that a graph approaches but never touches<\/li>\n<li>Use a function\u2019s derivatives to accurately sketch its graph<\/li>\n<\/ul>\n<\/section>\n<h2>Limits at Infinity<\/h2>\n<p>We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function [latex]f[\/latex] defined on an unbounded domain, we also need to know the behavior of [latex]f[\/latex] as [latex]x \\to \\pm \\infty[\/latex]. In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function [latex]f[\/latex].<\/p>\n<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.<\/p>\n<section class=\"textbox example\">\n<p>Consider the function [latex]f(x)=2+\\frac{1}{x}[\/latex].<\/p>\n<p>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 [latex]2[\/latex]. We say the limit as [latex]x[\/latex] approaches [latex]\\infty[\/latex] of [latex]f(x)[\/latex] is [latex]2[\/latex] and write:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}f(x)=2[\/latex].<\/p>\n<p>Similarly, for [latex]x<0[\/latex], as the values [latex]|x|[\/latex] get larger, the values of [latex]f(x)[\/latex] approaches [latex]2[\/latex]. We say the limit as [latex]x[\/latex] approaches [latex]\u2212\\infty[\/latex] of [latex]f(x)[\/latex] is [latex]2[\/latex] and write:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=2[\/latex].<\/p>\n<figure style=\"width: 717px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 1. The function approaches the asymptote [latex]y=2[\/latex] as [latex]x[\/latex] approaches [latex]\\pm \\infty[\/latex].<\/figcaption><\/figure>\n<table id=\"fs-id1165043428402\" class=\"column-header aligncenter\" 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>[latex]10[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<td>[latex]1,000[\/latex]<\/td>\n<td>[latex]10,000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]2+\\frac{1}{x}[\/latex]<\/strong><\/td>\n<td>[latex]2.1[\/latex]<\/td>\n<td>[latex]2.01[\/latex]<\/td>\n<td>[latex]2.001[\/latex]<\/td>\n<td>[latex]2.0001[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-10[\/latex]<\/td>\n<td>[latex]-100[\/latex]<\/td>\n<td>[latex]-1000[\/latex]<\/td>\n<td>[latex]-10,000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]2+\\frac{1}{x}[\/latex]<\/strong><\/td>\n<td>[latex]1.9[\/latex]<\/td>\n<td>[latex]1.99[\/latex]<\/td>\n<td>[latex]1.999[\/latex]<\/td>\n<td>[latex]1.9999[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<p>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:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to \\infty}{\\lim}f(x)=L[\/latex].<\/p>\n<p>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:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=L[\/latex].<\/p>\n<p>We now look at the definition of a function having a limit at infinity.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">limit at infinity (informal)<\/h3>\n<p id=\"fs-id1165042970725\">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<\/section>\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].<\/p>\n<section class=\"textbox example\">\n<p>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<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>horizontal asymptote<\/h3>\n<p>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<\/section>\n<figure style=\"width: 766px\" class=\"wp-caption alignnone\"><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\" \/><figcaption class=\"wp-caption-text\">Figure 2: 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 -\\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].<\/figcaption><\/figure>\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.<\/p>\n<section class=\"textbox example\">\n<p>The function [latex]f(x)=\\frac{ \\cos x}{x}+1[\/latex] shown below 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<figure style=\"width: 529px\" 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\" \/><figcaption 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.<\/figcaption><\/figure>\n<\/section>\n<p id=\"fs-id1165042373486\">The algebraic limit laws and squeeze theorem we introduced earlier\u00a0also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.<\/p>\n<section class=\"textbox example\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043183885\">Show Solution<\/button><\/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:\n<div style=\"text-align: center;\">[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].<\/div>\n<p>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.<\/p>\n<figure style=\"width: 492px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 4. This function approaches a horizontal asymptote as [latex]x\\to \\pm \\infty[\/latex].<\/figcaption><\/figure>\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<figure style=\"width: 717px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 5. This function crosses its horizontal asymptote multiple times.<\/figcaption><\/figure>\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.<br \/>\n<figure style=\"width: 492px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 6. The graph of [latex] \\tan x[\/latex] has vertical asymptotes at [latex]x=\\pm \\frac{\\pi }{2}[\/latex]<\/figcaption><\/figure>\n<\/li>\n<\/ol>\n<p id=\"fs-id1165043092430\">Since [latex]\\underset{x\\to (\\pi\/2)^-}{\\lim} \\tan x=\\infty[\/latex], 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 id=\"fs-id1165043097156\">Similarly, since [latex]\\underset{x\\to (\\pi\/2)^+}{\\lim} \\tan x=\u2212\\infty[\/latex], 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>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<figure style=\"width: 491px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 7. This function has two horizontal asymptotes.<\/figcaption><\/figure>\n<p>Watch the following video to see the worked solution to this 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=70&amp;end=307&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>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<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm169165\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=169165&theme=lumen&iframe_resize_id=ohm169165&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/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].<\/p>\n<p>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<section class=\"textbox example\">\n<p id=\"fs-id1165042606820\">Consider the function [latex]f(x)=x^3[\/latex].<\/p>\n<figure style=\"width: 642px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 8. For this function, the functional values approach infinity as [latex]x\\to \\pm \\infty[\/latex].<\/figcaption><\/figure>\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>[latex]10[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]50[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<td>[latex]1000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x^3[\/latex]<\/strong><\/td>\n<td>[latex]1000[\/latex]<\/td>\n<td>[latex]8000[\/latex]<\/td>\n<td>[latex]125,000[\/latex]<\/td>\n<td>[latex]1,000,000[\/latex]<\/td>\n<td>[latex]1,000,000,000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-10[\/latex]<\/td>\n<td>[latex]-20[\/latex]<\/td>\n<td>[latex]-50[\/latex]<\/td>\n<td>[latex]-100[\/latex]<\/td>\n<td>[latex]-1000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x^3[\/latex]<\/strong><\/td>\n<td>[latex]-1000[\/latex]<\/td>\n<td>[latex]-8000[\/latex]<\/td>\n<td>[latex]-125,000[\/latex]<\/td>\n<td>[latex]-1,000,000[\/latex]<\/td>\n<td>[latex]-1,000,000,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As seen in the table and figure above, 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].<\/p>\n<p>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<\/section>\n<section class=\"textbox keyTakeaway\">\n<div class=\"title\">\n<h3 style=\"text-align: left;\">infinite limits at infinity (informal)<\/h3>\n<\/div>\n<p id=\"fs-id1165043276356\">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.<\/p>\n<p>&nbsp;<\/p>\n<p>Similarly, we can define infinite limits as [latex]x\\to \u2212\\infty[\/latex].<\/p>\n<\/section>\n","protected":false},"author":6,"menu_order":5,"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":292,"module-header":"learn_it","content_attributions":[{"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":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/312"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/312\/revisions"}],"predecessor-version":[{"id":4546,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/312\/revisions\/4546"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/292"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/312\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=312"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=312"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=312"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}