{"id":313,"date":"2023-09-20T22:49:01","date_gmt":"2023-09-20T22:49:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/end-behavior\/"},"modified":"2024-08-05T02:24:32","modified_gmt":"2024-08-05T02:24:32","slug":"limits-at-infinity-and-asymptotes-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/limits-at-infinity-and-asymptotes-learn-it-3\/","title":{"raw":"Limits at Infinity and Asymptotes: Learn It 3","rendered":"Limits at Infinity and Asymptotes: Learn It 3"},"content":{"raw":"<h2>End Behavior<\/h2>\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<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<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 and Figure 12, we see that,<\/p>\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\n<p>and,<\/p>\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\r\n[caption id=\"\" align=\"alignleft\" 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] [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]\r\n\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.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>evaluating limits of power functions<\/h3>\r\n<p>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<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<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<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<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<\/section>\r\n<section class=\"textbox example\">\r\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>\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<p>[reveal-answer q=\"fs-id1165043254252\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043254252\"]<\/p>\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 [latex]-5[\/latex], 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 [latex]2[\/latex], 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<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<p><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\" data-mce-fragment=\"1\"><\/iframe><\/p>\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\/4.6LimitsAtInfinityAndAsymptotes877to983_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]<\/p>\r\n<\/section>\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].<\/p>\r\n<section class=\"textbox example\">\r\n<p>Consider a polynomial function<\/p>\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>\u00a0<\/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<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>\u00a0<\/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<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<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165043317360\">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 below.<\/p>\r\n\r\n[caption id=\"\" align=\"alignleft\" 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\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.\">\r\n<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>[latex]10[\/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]f(x)=5x^3-3x^2+4[\/latex]<\/strong><\/td>\r\n<td>[latex]4704[\/latex]<\/td>\r\n<td>[latex]4,970,004[\/latex]<\/td>\r\n<td>[latex]4,997,000,004[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]g(x)=5x^3[\/latex]<\/strong><\/td>\r\n<td>[latex]5000[\/latex]<\/td>\r\n<td>[latex]5,000,000[\/latex]<\/td>\r\n<td>[latex]5,000,000,000[\/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]-100[\/latex]<\/td>\r\n<td>[latex]-1000[\/latex]<\/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>[latex]-5296[\/latex]<\/td>\r\n<td>[latex]-5,029,996[\/latex]<\/td>\r\n<td>[latex]-5,002,999,996[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]g(x)=5x^3[\/latex]<\/strong><\/td>\r\n<td>[latex]-5000[\/latex]<\/td>\r\n<td>[latex]-5,000,000[\/latex]<\/td>\r\n<td>[latex]-5,000,000,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>","rendered":"<h2>End Behavior<\/h2>\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 and 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<figure style=\"width: 425px\" class=\"wp-caption alignleft\"><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\" \/><figcaption 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].<\/figcaption><\/figure>\n<figure style=\"width: 417px\" 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\" \/><figcaption 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].<\/figcaption><\/figure>\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.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>evaluating limits of power functions<\/h3>\n<p>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<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<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<\/section>\n<section class=\"textbox example\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043254252\">Show Solution<\/button><\/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 [latex]-5[\/latex], 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 [latex]2[\/latex], 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<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><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\" data-mce-fragment=\"1\"><\/iframe><\/p>\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\/4.6LimitsAtInfinityAndAsymptotes877to983_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<\/section>\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].<\/p>\n<section class=\"textbox example\">\n<p>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>\u00a0<\/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>\u00a0<\/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<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165043317360\">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 below.<\/p>\n<figure style=\"width: 382px\" class=\"wp-caption alignleft\"><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\" \/><figcaption class=\"wp-caption-text\">Figure 13. The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.<\/figcaption><\/figure>\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>[latex]10[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<td>[latex]1000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]f(x)=5x^3-3x^2+4[\/latex]<\/strong><\/td>\n<td>[latex]4704[\/latex]<\/td>\n<td>[latex]4,970,004[\/latex]<\/td>\n<td>[latex]4,997,000,004[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]g(x)=5x^3[\/latex]<\/strong><\/td>\n<td>[latex]5000[\/latex]<\/td>\n<td>[latex]5,000,000[\/latex]<\/td>\n<td>[latex]5,000,000,000[\/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]-100[\/latex]<\/td>\n<td>[latex]-1000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]f(x)=5x^3-3x^2+4[\/latex]<\/strong><\/td>\n<td>[latex]-5296[\/latex]<\/td>\n<td>[latex]-5,029,996[\/latex]<\/td>\n<td>[latex]-5,002,999,996[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]g(x)=5x^3[\/latex]<\/strong><\/td>\n<td>[latex]-5000[\/latex]<\/td>\n<td>[latex]-5,000,000[\/latex]<\/td>\n<td>[latex]-5,000,000,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n","protected":false},"author":6,"menu_order":7,"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\/313"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/313\/revisions"}],"predecessor-version":[{"id":3112,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/313\/revisions\/3112"}],"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\/313\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=313"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=313"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=313"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=313"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}