{"id":3086,"date":"2024-06-12T16:46:18","date_gmt":"2024-06-12T16:46:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3086"},"modified":"2025-08-17T23:56:15","modified_gmt":"2025-08-17T23:56:15","slug":"limits-at-infinity-and-asymptotes-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/limits-at-infinity-and-asymptotes-fresh-take\/","title":{"raw":"Limits at Infinity and Asymptotes: Fresh Take","rendered":"Limits at Infinity and Asymptotes: Fresh Take"},"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<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Limits at Infinity:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Describes behavior of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]\\infty[\/latex] or [latex]-\\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Notation: [latex]\\lim_{x \\to \\infty} f(x) = L[\/latex] or [latex]\\lim_{x \\to -\\infty} f(x) = L[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Horizontal Asymptotes:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Line [latex]y = L[\/latex] where [latex]f(x)[\/latex] approaches [latex]L[\/latex] as [latex]x \\to \\pm\\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Function may cross horizontal asymptote multiple times<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Infinite Limits at Infinity:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">When [latex]f(x)[\/latex] grows without bound as [latex]x \\to \\pm\\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Notation: [latex]\\lim_{x \\to \\infty} f(x) = \\infty[\/latex] or [latex]\\lim_{x \\to \\infty} f(x) = -\\infty[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Formal Definitions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For finite limits: [latex]\\forall \\varepsilon &gt; 0, \\exists N &gt; 0: |f(x) - L| &lt; \\varepsilon[\/latex] when [latex]x &gt; N[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For infinite limits: [latex]\\forall M &gt; 0, \\exists N &gt; 0: f(x) &gt; M[\/latex] when [latex]x &gt; N[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Strategies for Evaluation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Use algebraic limit laws<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply the squeeze theorem<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Analyze rational functions by comparing highest degree terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165043315935\">Evaluate [latex]\\underset{x\\to \u2212\\infty}{\\lim}\\left(3+\\frac{4}{x}\\right)[\/latex] and [latex]\\underset{x\\to \\infty }{\\lim}\\left(3+\\frac{4}{x}\\right)[\/latex]. Determine the horizontal asymptotes of [latex]f(x)=3+\\frac{4}{x}[\/latex], if any.<\/p>\r\n<p>[reveal-answer q=\"2473508\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"2473508\"]<\/p>\r\n<p id=\"fs-id1165042318511\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}1\/x=0[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043390798\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043390798\"]<\/p>\r\n<p id=\"fs-id1165043390798\">Both limits are [latex]3[\/latex]. The line [latex]y=3[\/latex] is a horizontal asymptote.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165042480099\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}\\left(3 - \\dfrac{1}{x^2}\\right)=3[\/latex].<\/p>\r\n<p>[reveal-answer q=\"9822541\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"9822541\"]<\/p>\r\n<p id=\"fs-id1165042332065\">Let [latex]N=\\dfrac{1}{\\sqrt{\\varepsilon }}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042367887\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042367887\"]<\/p>\r\n<p id=\"fs-id1165042367887\">Let [latex]\\varepsilon &gt;0[\/latex]. Let [latex]N=\\dfrac{1}{\\sqrt{\\varepsilon }}[\/latex]. Therefore, for all [latex]x&gt;N[\/latex], we have<\/p>\r\n<p id=\"fs-id1165042376362\">[latex]|3-\\dfrac{1}{x^2}-3|=\\dfrac{1}{x^2}&lt;\\dfrac{1}{N^2}=\\varepsilon [\/latex]<\/p>\r\n<p id=\"fs-id1165042320298\">Therefore, [latex]\\underset{x\\to \\infty }{\\lim}\\left(3-\\frac{1}{x^2}\\right)=3[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165042320226\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}3x^2=\\infty[\/latex].<\/p>\r\n<p>[reveal-answer q=\"80775166\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"80775166\"]<\/p>\r\n<p id=\"fs-id1165043219104\">Let [latex]N=\\sqrt{\\frac{M}{3}}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042708272\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042708272\"]<\/p>\r\n<p id=\"fs-id1165042708272\">Let [latex]M&gt;0[\/latex]. Let [latex]N=\\sqrt{\\frac{M}{3}}[\/latex]. Then, for all [latex]x&gt;N[\/latex], we have,<\/p>\r\n<p id=\"fs-id1165042383154\">[latex]3x^2&gt;3N^2=3(\\sqrt{\\frac{M}{3}})^2=\\frac{3M}{3}=M[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>End Behavior<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">End Behavior:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Describes how a function behaves as [latex]x \\to \\pm\\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Three possible outcomes: approach a finite value, approach infinity, or oscillate<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Polynomial End Behavior:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Determined by the highest degree term<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Even degree: same behavior at both ends<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Odd degree: opposite behavior at each end<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Rational Function End Behavior:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Depends on the degree relationship between numerator and denominator<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Three cases: horizontal asymptote, zero asymptote, or unbounded growth<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Horizontal Asymptotes:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]y = L[\/latex] where [latex]\\lim_{x \\to \\pm\\infty} f(x) = L[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For rational functions [latex]\\frac{p(x)}{q(x)}[\/latex]:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If deg(p) &lt; deg(q): [latex]y = 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">If deg(p) = deg(q): [latex]y = \\frac{a_n}{b_m}[\/latex] (ratio of leading coefficients)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">If deg(p) &gt; deg(q): no horizontal asymptote<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Transcendental Function End Behavior:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Trigonometric: oscillate (no limit)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Exponential ([latex]e^x[\/latex]): [latex]\\to 0[\/latex] as [latex]x \\to -\\infty[\/latex], [latex]\\to \\infty[\/latex] as [latex]x \\to \\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Natural log: [latex]\\to -\\infty[\/latex] as [latex]x \\to 0^+[\/latex], [latex]\\to \\infty[\/latex] as [latex]x \\to \\infty[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\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<p>[reveal-answer q=\"2288937\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"2288937\"]<\/p>\r\n<p id=\"fs-id1165042708228\">The coefficient [latex]-3[\/latex] is negative.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042708212\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042708212\"]<\/p>\r\n<p id=\"fs-id1165042708212\">[latex]\u2212\\infty [\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\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<p>[reveal-answer q=\"4338920\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4338920\"]<\/p>\r\n<p id=\"fs-id1165042374889\">Divide the numerator and denominator by [latex]x^2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042374871\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042374871\"]<\/p>\r\n<p id=\"fs-id1165042374871\">[latex]\\frac{3}{5}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165042463923\">Evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\sqrt{3x^2+4}}{x+6}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"1990672\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"1990672\"]<\/p>\r\n<p id=\"fs-id1165043327311\">Divide the numerator and denominator by [latex]|x|[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043327293\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043327293\"]<\/p>\r\n<p id=\"fs-id1165043327293\">[latex]\\pm \\sqrt{3}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\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<p>[reveal-answer q=\"377625\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"377625\"]<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042711402\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042711402\"]<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Drawing Graphs of Functions<\/h2>\r\n<section class=\"textbox example\">\r\n<p>Sketch a graph of [latex]f(x)=(x-1)^3 (x+2)[\/latex]<\/p>\r\n<div id=\"fs-id1165042710050\" class=\"exercise\">\r\n<p>[reveal-answer q=\"80046723\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"80046723\"]<\/p>\r\n<p id=\"fs-id1165042710130\">[latex]f[\/latex] is a fourth-degree polynomial.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042710106\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042710106\"]<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"401\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211136\/CNX_Calc_Figure_04_06_028.jpg\" alt=\"The function f(x) = (x \u22121)3(x + 2) is graphed.\" width=\"401\" height=\"520\" \/> Figure 24. Graph of f(x)[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Sketch a graph of [latex]f(x)=\\dfrac{3x+5}{8+4x}[\/latex]<\/p>\r\n<div id=\"fs-id1165042592768\" class=\"exercise\">\r\n<p>[reveal-answer q=\"3127001\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3127001\"]<\/p>\r\n<p id=\"fs-id1165042592850\">A line [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex] if the limit as [latex]x\\to \\infty [\/latex] or the limit as [latex]x\\to \u2212\\infty [\/latex] of [latex]f(x)[\/latex] is [latex]L[\/latex]. A line [latex]x=a[\/latex] is a vertical asymptote if at least one of the one-sided limits of [latex]f[\/latex] as [latex]x\\to a[\/latex] is [latex]\\infty [\/latex] or [latex]\u2212\\infty[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042592825\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042592825\"]<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"717\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211142\/CNX_Calc_Figure_04_06_029.jpg\" alt=\"The function f(x) = (3x + 5)\/(8 + 4x) is graphed. It appears to have asymptotes at x = \u22122 and y = 1.\" width=\"717\" height=\"422\" \/> Figure 26. Graph of f(x)[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Find the oblique asymptote for [latex]f(x)=\\dfrac{3x^3-2x+1}{2x^2-4}[\/latex]<\/p>\r\n\r\n[reveal-answer q=\"99083451\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"99083451\"]\r\n\r\n<p id=\"fs-id1165042607084\">Use long division of polynomials.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p id=\"fs-id1165042607059\">[reveal-answer q=\"fs-id1165042607059\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042607059\"]<\/p>\r\n<p>[latex]y=\\frac{3}{2}x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Consider the function [latex]f(x)=5-x^{\\frac{2}{3}}[\/latex]. Determine the point on the graph where a cusp is located. Determine the end behavior of [latex]f[\/latex].<\/p>\r\n<p>[reveal-answer q=\"44668822\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"44668822\"]<\/p>\r\n<p>A function [latex]f[\/latex] has a cusp at a point [latex]a[\/latex] if [latex]f(a)[\/latex] exists, [latex]f^{\\prime}(a)[\/latex] is undefined, one of the one-sided limits as [latex]x\\to a[\/latex] of [latex]f^{\\prime}(x)[\/latex] is [latex]+\\infty[\/latex], and the other one-sided limit is [latex]\u2212\\infty[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042644060\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042644060\"]<\/p>\r\n<p>The function [latex]f[\/latex] has a cusp at [latex](0,5)[\/latex]: [latex]\\underset{x\\to 0^-}{\\lim}f^{\\prime}(x)=\\infty[\/latex], [latex]\\underset{x\\to 0^+}{\\lim}f^{\\prime}(x)=\u2212\\infty[\/latex]. For end behavior, [latex]\\underset{x\\to \\pm \\infty }{\\lim}f(x)=\u2212\\infty[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/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<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Limits at Infinity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Describes behavior of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]\\infty[\/latex] or [latex]-\\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Notation: [latex]\\lim_{x \\to \\infty} f(x) = L[\/latex] or [latex]\\lim_{x \\to -\\infty} f(x) = L[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Line [latex]y = L[\/latex] where [latex]f(x)[\/latex] approaches [latex]L[\/latex] as [latex]x \\to \\pm\\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Function may cross horizontal asymptote multiple times<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Infinite Limits at Infinity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">When [latex]f(x)[\/latex] grows without bound as [latex]x \\to \\pm\\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Notation: [latex]\\lim_{x \\to \\infty} f(x) = \\infty[\/latex] or [latex]\\lim_{x \\to \\infty} f(x) = -\\infty[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Formal Definitions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For finite limits: [latex]\\forall \\varepsilon > 0, \\exists N > 0: |f(x) - L| < \\varepsilon[\/latex] when [latex]x > N[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For infinite limits: [latex]\\forall M > 0, \\exists N > 0: f(x) > M[\/latex] when [latex]x > N[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Strategies for Evaluation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use algebraic limit laws<\/li>\n<li class=\"whitespace-normal break-words\">Apply the squeeze theorem<\/li>\n<li class=\"whitespace-normal break-words\">Analyze rational functions by comparing highest degree terms<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1165043315935\">Evaluate [latex]\\underset{x\\to \u2212\\infty}{\\lim}\\left(3+\\frac{4}{x}\\right)[\/latex] and [latex]\\underset{x\\to \\infty }{\\lim}\\left(3+\\frac{4}{x}\\right)[\/latex]. Determine the horizontal asymptotes of [latex]f(x)=3+\\frac{4}{x}[\/latex], if any.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q2473508\">Hint<\/button><\/p>\n<div id=\"q2473508\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042318511\">[latex]\\underset{x\\to \\pm \\infty }{\\lim}1\/x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043390798\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043390798\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043390798\">Both limits are [latex]3[\/latex]. The line [latex]y=3[\/latex] is a horizontal asymptote.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165042480099\">Use the formal definition of limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}\\left(3 - \\dfrac{1}{x^2}\\right)=3[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q9822541\">Hint<\/button><\/p>\n<div id=\"q9822541\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042332065\">Let [latex]N=\\dfrac{1}{\\sqrt{\\varepsilon }}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042367887\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042367887\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042367887\">Let [latex]\\varepsilon >0[\/latex]. Let [latex]N=\\dfrac{1}{\\sqrt{\\varepsilon }}[\/latex]. Therefore, for all [latex]x>N[\/latex], we have<\/p>\n<p id=\"fs-id1165042376362\">[latex]|3-\\dfrac{1}{x^2}-3|=\\dfrac{1}{x^2}<\\dfrac{1}{N^2}=\\varepsilon[\/latex]<\/p>\n<p id=\"fs-id1165042320298\">Therefore, [latex]\\underset{x\\to \\infty }{\\lim}\\left(3-\\frac{1}{x^2}\\right)=3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165042320226\">Use the formal definition of infinite limit at infinity to prove that [latex]\\underset{x\\to \\infty }{\\lim}3x^2=\\infty[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q80775166\">Hint<\/button><\/p>\n<div id=\"q80775166\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043219104\">Let [latex]N=\\sqrt{\\frac{M}{3}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042708272\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042708272\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042708272\">Let [latex]M>0[\/latex]. Let [latex]N=\\sqrt{\\frac{M}{3}}[\/latex]. Then, for all [latex]x>N[\/latex], we have,<\/p>\n<p id=\"fs-id1165042383154\">[latex]3x^2>3N^2=3(\\sqrt{\\frac{M}{3}})^2=\\frac{3M}{3}=M[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>End Behavior<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">End Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Describes how a function behaves as [latex]x \\to \\pm\\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Three possible outcomes: approach a finite value, approach infinity, or oscillate<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Polynomial End Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determined by the highest degree term<\/li>\n<li class=\"whitespace-normal break-words\">Even degree: same behavior at both ends<\/li>\n<li class=\"whitespace-normal break-words\">Odd degree: opposite behavior at each end<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Rational Function End Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Depends on the degree relationship between numerator and denominator<\/li>\n<li class=\"whitespace-normal break-words\">Three cases: horizontal asymptote, zero asymptote, or unbounded growth<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]y = L[\/latex] where [latex]\\lim_{x \\to \\pm\\infty} f(x) = L[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For rational functions [latex]\\frac{p(x)}{q(x)}[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If deg(p) &lt; deg(q): [latex]y = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If deg(p) = deg(q): [latex]y = \\frac{a_n}{b_m}[\/latex] (ratio of leading coefficients)<\/li>\n<li class=\"whitespace-normal break-words\">If deg(p) &gt; deg(q): no horizontal asymptote<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Transcendental Function End Behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Trigonometric: oscillate (no limit)<\/li>\n<li class=\"whitespace-normal break-words\">Exponential ([latex]e^x[\/latex]): [latex]\\to 0[\/latex] as [latex]x \\to -\\infty[\/latex], [latex]\\to \\infty[\/latex] as [latex]x \\to \\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Natural log: [latex]\\to -\\infty[\/latex] as [latex]x \\to 0^+[\/latex], [latex]\\to \\infty[\/latex] as [latex]x \\to \\infty[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\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><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q2288937\">Hint<\/button><\/p>\n<div id=\"q2288937\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042708228\">The coefficient [latex]-3[\/latex] is negative.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042708212\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042708212\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042708212\">[latex]\u2212\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\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><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4338920\">Hint<\/button><\/p>\n<div id=\"q4338920\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042374889\">Divide the numerator and denominator by [latex]x^2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042374871\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042374871\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042374871\">[latex]\\frac{3}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165042463923\">Evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\sqrt{3x^2+4}}{x+6}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q1990672\">Hint<\/button><\/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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043327293\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q377625\">Hint<\/button><\/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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042711402\">Show Solution<\/button><\/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<\/section>\n<h2>Drawing Graphs of Functions<\/h2>\n<section class=\"textbox example\">\n<p>Sketch a graph of [latex]f(x)=(x-1)^3 (x+2)[\/latex]<\/p>\n<div id=\"fs-id1165042710050\" class=\"exercise\">\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q80046723\">Hint<\/button><\/p>\n<div id=\"q80046723\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042710130\">[latex]f[\/latex] is a fourth-degree polynomial.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042710106\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042710106\" class=\"hidden-answer\" style=\"display: none\">\n<figure style=\"width: 401px\" 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\/11211136\/CNX_Calc_Figure_04_06_028.jpg\" alt=\"The function f(x) = (x \u22121)3(x + 2) is graphed.\" width=\"401\" height=\"520\" \/><figcaption class=\"wp-caption-text\">Figure 24. Graph of f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Sketch a graph of [latex]f(x)=\\dfrac{3x+5}{8+4x}[\/latex]<\/p>\n<div id=\"fs-id1165042592768\" class=\"exercise\">\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3127001\">Hint<\/button><\/p>\n<div id=\"q3127001\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042592850\">A line [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex] if the limit as [latex]x\\to \\infty[\/latex] or the limit as [latex]x\\to \u2212\\infty[\/latex] of [latex]f(x)[\/latex] is [latex]L[\/latex]. A line [latex]x=a[\/latex] is a vertical asymptote if at least one of the one-sided limits of [latex]f[\/latex] as [latex]x\\to a[\/latex] is [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042592825\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042592825\" class=\"hidden-answer\" style=\"display: none\">\n<figure style=\"width: 717px\" 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\/11211142\/CNX_Calc_Figure_04_06_029.jpg\" alt=\"The function f(x) = (3x + 5)\/(8 + 4x) is graphed. It appears to have asymptotes at x = \u22122 and y = 1.\" width=\"717\" height=\"422\" \/><figcaption class=\"wp-caption-text\">Figure 26. Graph of f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the oblique asymptote for [latex]f(x)=\\dfrac{3x^3-2x+1}{2x^2-4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q99083451\">Hint<\/button><\/p>\n<div id=\"q99083451\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042607084\">Use long division of polynomials.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042607059\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042607059\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042607059\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y=\\frac{3}{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Consider the function [latex]f(x)=5-x^{\\frac{2}{3}}[\/latex]. Determine the point on the graph where a cusp is located. Determine the end behavior of [latex]f[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44668822\">Hint<\/button><\/p>\n<div id=\"q44668822\" class=\"hidden-answer\" style=\"display: none\">\n<p>A function [latex]f[\/latex] has a cusp at a point [latex]a[\/latex] if [latex]f(a)[\/latex] exists, [latex]f^{\\prime}(a)[\/latex] is undefined, one of the one-sided limits as [latex]x\\to a[\/latex] of [latex]f^{\\prime}(x)[\/latex] is [latex]+\\infty[\/latex], and the other one-sided limit is [latex]\u2212\\infty[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042644060\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042644060\" class=\"hidden-answer\" style=\"display: none\">\n<p>The function [latex]f[\/latex] has a cusp at [latex](0,5)[\/latex]: [latex]\\underset{x\\to 0^-}{\\lim}f^{\\prime}(x)=\\infty[\/latex], [latex]\\underset{x\\to 0^+}{\\lim}f^{\\prime}(x)=\u2212\\infty[\/latex]. For end behavior, [latex]\\underset{x\\to \\pm \\infty }{\\lim}f(x)=\u2212\\infty[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":292,"module-header":"fresh_take","content_attributions":[],"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\/3086"}],"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\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3086\/revisions"}],"predecessor-version":[{"id":4783,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3086\/revisions\/4783"}],"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\/3086\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3086"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3086"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3086"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3086"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}