{"id":1300,"date":"2024-04-16T12:56:22","date_gmt":"2024-04-16T12:56:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1300"},"modified":"2025-08-17T16:11:07","modified_gmt":"2025-08-17T16:11:07","slug":"understanding-limits-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/understanding-limits-background-youll-need-2\/","title":{"raw":"Understanding Limits: Background You\u2019ll Need 2","rendered":"Understanding Limits: Background You\u2019ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify vertical and horizontal asymptotes&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4995,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:16776960},&quot;10&quot;:2,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri, Arial&quot;}\">Find the vertical and horizontal asymptotes of a function<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Vertical Asymptote<\/h2>\r\n<p>It's important to recognize characteristics of a function's graph that signal specific behaviors. <strong>Vertical asymptotes<\/strong> are one such feature, indicating where a function's output heads towards infinity or negative infinity as the input nears certain values.\u00a0<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>vertical asymptote<\/strong><\/h3>\r\n<p>Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a certain value. They represent values of [latex]x[\/latex] at which the function is undefined and the graph of the function cannot cross. These asymptotes are typically found in rational functions where the denominator is zero.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213925\/CNX_Precalc_Figure_03_07_0092.jpg\" alt=\"Graph of f(x)=1\/(x-3) with its vertical asymptote at x=3 and its horizontal asymptote at y=0.\" width=\"487\" height=\"364\" \/> Graph of f(x) with a vertical and horizontal asymptote labeled[\/caption]\r\n\r\n<p>The graph of [latex]f\\left(x\\right)=\\dfrac{x+3}{{x}^{2}-9}[\/latex] has a vertical asymptote at [latex]x=3[\/latex].<\/p>\r\n<\/section>\r\n<p>In rational functions, vertical asymptotes serve as crucial indicators of the points at which the function's value tends toward infinity. These asymptotes can be found by examining the factors of the denominator that do not cancel with corresponding factors in the numerator. Specifically, the values that make the denominator zero\u2014unless they also make the numerator zero\u2014determine the locations of these asymptotes.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Given a Rational Function, Identify it's Vertical Asymptotes.<\/strong><\/p>\r\n<ol>\r\n\t<li><strong>Factor<\/strong>: Break down both the numerator and denominator into their prime factors.<\/li>\r\n\t<li><strong>Simplify<\/strong>: Reduce the rational function by canceling out common factors between the numerator and the denominator.<\/li>\r\n\t<li><strong>Determine Asymptotes<\/strong>: The zeros of the simplified denominator are the [latex]x[\/latex]-values for vertical asymptotes, provided they do not also zero the numerator.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<p>To break down an expression into prime factors, repeatedly divide by the smallest prime number that will go into the number evenly, until you reach a quotient of one. For algebraic expressions, apply factoring techniques such as finding common factors, difference of squares, or using the quadratic formula to express the expression as a product of its irreducible factors.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Find the vertical asymptotes of the graph of [latex]k\\left(x\\right)=\\dfrac{5+2{x}^{2}}{2-x-{x}^{2}}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"787718\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"787718\"]<\/p>\r\n<p>First, factor the numerator and denominator.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k\\left(x\\right)&amp;=\\dfrac{5+2{x}^{2}}{2-x-{x}^{2}} \\\\[1mm] &amp;=\\dfrac{5+2{x}^{2}}{\\left(2+x\\right)\\left(1-x\\right)} \\end{align}[\/latex]<\/p>\r\n<p>To find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to zero:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(2+x\\right)\\left(1-x\\right)=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=-2,1[\/latex]<\/p>\r\n<p>Neither [latex]x=-2[\/latex] nor [latex]x=1[\/latex] are zeros of the numerator, so the two values indicate two vertical asymptotes. The graph below confirms the location of the two vertical asymptotes.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213927\/CNX_Precalc_Figure_03_07_0102.jpg\" alt=\"Graph of k(x)=(5+2x)^2\/(2-x-x^2) with its vertical asymptotes at x=-2 and x=1 and its horizontal asymptote at y=-2.\" width=\"487\" height=\"514\" \/> Graph of k(x) with two vertical asymptotes and one horizontal asymptote labeled[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Horizontal Asymptote<\/h2>\r\n<p>While vertical asymptotes describe the behavior of a graph as the\u00a0<em data-effect=\"italics\">output<\/em>\u00a0gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the\u00a0<em data-effect=\"italics\">input<\/em>\u00a0gets very large or very small. Recall that a polynomial\u2019s end behavior will mirror that of the leading term. Likewise, a rational function\u2019s end behavior will mirror that of the ratio of the function that is the ratio of the leading terms.<\/p>\r\n<section class=\"textbox recall\">\r\n<p>The leading coefficient and degree of a polynomial function is useful when predicting its end behavior.<\/p>\r\n<p>If the leading term is positive or negative, and has even or odd degree, it will tell us the toolkit function's graph behavior it will mimic: [latex]f(x)=x^2, \\quad f(x)=-x^2,\\quad f(x)=x^3,\\quad[\/latex] or [latex]\\quad f(x)=-x^3[\/latex].<\/p>\r\n<p>The same idea applies to <em>the ratio of leading terms\u00a0<\/em>of a\u00a0rational function.<\/p>\r\n<\/section>\r\n<p>There are three distinct outcomes when checking for horizontal asymptotes:<\/p>\r\n<p><strong>Case 1:<\/strong> If the degree of the denominator is greater than the degree of the numerator, there is a <strong>horizontal asymptote<\/strong> at [latex]y=0[\/latex].<\/p>\r\n<section class=\"textbox example\">\r\n<p>Consider the following function:<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{4x+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\r\n<p>In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{4x}{{x}^{2}}=\\frac{4}{x}[\/latex].<\/p>\r\n<p>This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\\left(x\\right)=\\frac{4}{x}[\/latex], and the outputs will approach zero, resulting in a horizontal asymptote at [latex]y=0[\/latex]. Note that this graph crosses the horizontal asymptote.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213933\/CNX_Precalc_Figure_03_07_0132.jpg\" alt=\"Graph of f(x)=(4x+2)\/(x^2+4x-5) with its vertical asymptotes at x=-5 and x=1 and its horizontal asymptote at y=0.\" width=\"900\" height=\"302\" \/> Horizontal Asymptote [latex]y=0[\/latex] when [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne{0}\\text{ where degree of }p&lt;\\text{degree of q}[\/latex].[\/caption]\r\n<\/section>\r\n<p><strong>Case 2:<\/strong> If the degree of the denominator is less than the degree of the numerator by one, we get a <strong>slant asymptote<\/strong>.<\/p>\r\n<section class=\"textbox example\">\r\n<p>Consider the following function:<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{3{x}^{2}-2x+1}{x - 1}[\/latex]<\/p>\r\n<p>In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{x}=3x[\/latex].<\/p>\r\n<p>This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\\left(x\\right)=3x[\/latex]. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote.<\/p>\r\n<p>However, the graph of [latex]g\\left(x\\right)=3x[\/latex] looks like a diagonal line, and since [latex]f[\/latex] will behave similarly to [latex]g[\/latex], it will approach a line close to [latex]y=3x[\/latex]. This line is a slant asymptote.<\/p>\r\n<p>To find the equation of the slant asymptote, divide [latex]\\dfrac{3{x}^{2}-2x+1}{x - 1}[\/latex]. The quotient is [latex]3x+1[\/latex] and the remainder is [latex]2[\/latex]. The slant asymptote is the graph of the line [latex]g(x)=3x+1[\/latex].<\/p>\r\n\r\n[caption id=\"attachment_1823\" align=\"aligncenter\" width=\"532\"]<img class=\"wp-image-1823 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29160914\/Screenshot-2024-04-29-120903.png\" alt=\"Graph of f(x)=(3x^2-2x+1)\/(x-1) with its vertical asymptote at x=1 and a slant asymptote aty=3x+1.\" width=\"532\" height=\"539\" \/> Slant asymptote when [latex]f(x)=\\frac{p(x)}{q(x)}, q(x) \\neq 0[\/latex] where degree of [latex] p &gt; \\text{ degree of } q \\text{ by } 1[\/latex][\/caption]\r\n<\/section>\r\n<p><strong>Case 3:<\/strong> If the degree of the denominator is equal to the degree of the numerator, there is a horizontal asymptote at [latex]y=\\frac{{a}_{n}}{{b}_{n}}[\/latex], where [latex]{a}_{n}[\/latex] and [latex]{b}_{n}[\/latex] are the leading coefficients of [latex]p\\left(x\\right)[\/latex] and [latex]q\\left(x\\right)[\/latex] for [latex]f\\left(x\\right)=\\frac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0[\/latex].<\/p>\r\n<section class=\"textbox example\">\r\n<p>Consider the following function:<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{3{x}^{2}+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{{x}^{2}}=3[\/latex].<\/p>\r\n<p style=\"text-align: left;\">This tells us that as the inputs grow large, this function will behave like the function [latex]g\\left(x\\right)=3[\/latex], which is a horizontal line. As [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to 3[\/latex], resulting in a horizontal asymptote at [latex]y=3[\/latex]. Note that this graph crosses the horizontal asymptote.<\/p>\r\n\r\n[caption id=\"attachment_1826\" align=\"alignnone\" width=\"802\"]<img class=\"size-full wp-image-1826\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29161246\/Screenshot-2024-04-29-121237.png\" alt=\"Graph of f(x)=(3x^2+2)\/(x^2+4x-5) with its vertical asymptotes at x=-5 and x=1 and its horizontal asymptote at y=3.\" width=\"802\" height=\"374\" \/> Horizontal Asymptote when [latex]f\\left(x\\right)=\\frac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0\\text{ where degree of }p=\\text{degree of }q[\/latex].[\/caption]\r\n\r\n<p>&nbsp;<\/p>\r\n<\/section>\r\n<p>Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.<\/p>\r\n<section class=\"textbox proTip\">\r\n<p>It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction.<\/p>\r\n<p>For instance, if we had the function<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{3{x}^{5}-{x}^{2}}{x+3}[\/latex]<\/p>\r\n<p>with end behavior<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)\\approx \\dfrac{3{x}^{5}}{x}=3{x}^{4}[\/latex],<\/p>\r\n<p>the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.<\/p>\r\n<p style=\"text-align: center;\">As [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to \\infty [\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>horizontal asymptote<\/strong><\/h3>\r\n<p>The <strong>horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.<\/p>\r\n<ul id=\"fs-id1165137722720\">\r\n\t<li><strong>Case 1:<\/strong> Degree of denominator [latex]&gt;[\/latex] degree of\u00a0 the numerator: horizontal asymptote at [latex]y=0[\/latex]<\/li>\r\n\t<li><strong>Case 2<\/strong>: Degree of denominator [latex]&lt;[\/latex] degree of numerator by one: no horizontal asymptote; slant asymptote.\r\n\r\n<ul>\r\n\t<li>If the degree of the numerator is greater than the degree of the denominator by\u00a0<em>more than one<\/em>, the end behavior of the function's graph will mimic that of the graph of the reduced ratio of leading terms.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li><strong>Case 3<\/strong>: Degree of numerator [latex]=[\/latex]\u00a0degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>For the functions below, identify the horizontal or slant asymptote.<\/p>\r\n<ol>\r\n\t<li>[latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]<\/li>\r\n\t<li>[latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]<\/li>\r\n\t<li>[latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"968793\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"968793\"]<\/p>\r\n<p>For these solutions, we will use [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)}, q\\left(x\\right)\\ne 0[\/latex].<\/p>\r\n<ol>\r\n\t<li>\r\n<p>The degree of [latex]p[\/latex] and the degree of [latex]q[\/latex] are both equal to [latex]3[\/latex], so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at [latex]y=\\frac{6}{2}[\/latex] or [latex]y=3[\/latex].<\/p>\r\n<\/li>\r\n\t<li>\r\n<p>The degree of [latex]p=2[\/latex] and degree of [latex]q=1[\/latex]. Since [latex]p&gt;q[\/latex] by [latex]1[\/latex], there is a slant asymptote found at [latex]\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex].<\/p>\r\n<center>\r\n[caption id=\"attachment_4515\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-4515 size-medium\" style=\"text-align: center;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/21031600\/CNX_CAT_Figure_05_01_011-300x71.jpg\" alt=\"Synthetic division of x^2-4x+1 by x+2, resulting in x-6 with a remainder of 13\" width=\"300\" height=\"71\" \/> Synthetic division[\/caption]\r\n<\/center>\r\n<p>The quotient is [latex]x - 6[\/latex] and the remainder is [latex]13[\/latex]. There is a slant asymptote at [latex]y=-x - 6[\/latex].<\/p>\r\n<\/li>\r\n\t<li>\r\n<p>The degree of [latex]p=2\\text{ }&lt;[\/latex] degree of [latex]q=3[\/latex], so there is a horizontal asymptote [latex]y=0[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288268[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Find the horizontal and vertical asymptotes of the function<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{\\left(x - 2\\right)\\left(x+3\\right)}{\\left(x - 1\\right)\\left(x+2\\right)\\left(x - 5\\right)}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"228875\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"228875\"]<\/p>\r\n<p>First, note that this function has no common factors, so there are no potential removable discontinuities.<\/p>\r\n<p>The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at [latex]x=1,-2,\\text{and }5[\/latex], indicating vertical asymptotes at these values.<\/p>\r\n<p>The numerator has degree [latex]2[\/latex], while the denominator has degree [latex]3[\/latex]. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to 0[\/latex]. This function will have a horizontal asymptote at [latex]y=0[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]105058[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Identify vertical and horizontal asymptotes&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4995,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:16776960},&quot;10&quot;:2,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri, Arial&quot;}\">Find the vertical and horizontal asymptotes of a function<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Vertical Asymptote<\/h2>\n<p>It&#8217;s important to recognize characteristics of a function&#8217;s graph that signal specific behaviors. <strong>Vertical asymptotes<\/strong> are one such feature, indicating where a function&#8217;s output heads towards infinity or negative infinity as the input nears certain values.\u00a0<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>vertical asymptote<\/strong><\/h3>\n<p>Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a certain value. They represent values of [latex]x[\/latex] at which the function is undefined and the graph of the function cannot cross. These asymptotes are typically found in rational functions where the denominator is zero.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213925\/CNX_Precalc_Figure_03_07_0092.jpg\" alt=\"Graph of f(x)=1\/(x-3) with its vertical asymptote at x=3 and its horizontal asymptote at y=0.\" width=\"487\" height=\"364\" \/><figcaption class=\"wp-caption-text\">Graph of f(x) with a vertical and horizontal asymptote labeled<\/figcaption><\/figure>\n<p>The graph of [latex]f\\left(x\\right)=\\dfrac{x+3}{{x}^{2}-9}[\/latex] has a vertical asymptote at [latex]x=3[\/latex].<\/p>\n<\/section>\n<p>In rational functions, vertical asymptotes serve as crucial indicators of the points at which the function&#8217;s value tends toward infinity. These asymptotes can be found by examining the factors of the denominator that do not cancel with corresponding factors in the numerator. Specifically, the values that make the denominator zero\u2014unless they also make the numerator zero\u2014determine the locations of these asymptotes.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Given a Rational Function, Identify it&#8217;s Vertical Asymptotes.<\/strong><\/p>\n<ol>\n<li><strong>Factor<\/strong>: Break down both the numerator and denominator into their prime factors.<\/li>\n<li><strong>Simplify<\/strong>: Reduce the rational function by canceling out common factors between the numerator and the denominator.<\/li>\n<li><strong>Determine Asymptotes<\/strong>: The zeros of the simplified denominator are the [latex]x[\/latex]-values for vertical asymptotes, provided they do not also zero the numerator.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox recall\">\n<p>To break down an expression into prime factors, repeatedly divide by the smallest prime number that will go into the number evenly, until you reach a quotient of one. For algebraic expressions, apply factoring techniques such as finding common factors, difference of squares, or using the quadratic formula to express the expression as a product of its irreducible factors.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the vertical asymptotes of the graph of [latex]k\\left(x\\right)=\\dfrac{5+2{x}^{2}}{2-x-{x}^{2}}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q787718\">Show Solution<\/button><\/p>\n<div id=\"q787718\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, factor the numerator and denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k\\left(x\\right)&=\\dfrac{5+2{x}^{2}}{2-x-{x}^{2}} \\\\[1mm] &=\\dfrac{5+2{x}^{2}}{\\left(2+x\\right)\\left(1-x\\right)} \\end{align}[\/latex]<\/p>\n<p>To find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to zero:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2+x\\right)\\left(1-x\\right)=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=-2,1[\/latex]<\/p>\n<p>Neither [latex]x=-2[\/latex] nor [latex]x=1[\/latex] are zeros of the numerator, so the two values indicate two vertical asymptotes. The graph below confirms the location of the two vertical asymptotes.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213927\/CNX_Precalc_Figure_03_07_0102.jpg\" alt=\"Graph of k(x)=(5+2x)^2\/(2-x-x^2) with its vertical asymptotes at x=-2 and x=1 and its horizontal asymptote at y=-2.\" width=\"487\" height=\"514\" \/><figcaption class=\"wp-caption-text\">Graph of k(x) with two vertical asymptotes and one horizontal asymptote labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<h2>Horizontal Asymptote<\/h2>\n<p>While vertical asymptotes describe the behavior of a graph as the\u00a0<em data-effect=\"italics\">output<\/em>\u00a0gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the\u00a0<em data-effect=\"italics\">input<\/em>\u00a0gets very large or very small. Recall that a polynomial\u2019s end behavior will mirror that of the leading term. Likewise, a rational function\u2019s end behavior will mirror that of the ratio of the function that is the ratio of the leading terms.<\/p>\n<section class=\"textbox recall\">\n<p>The leading coefficient and degree of a polynomial function is useful when predicting its end behavior.<\/p>\n<p>If the leading term is positive or negative, and has even or odd degree, it will tell us the toolkit function&#8217;s graph behavior it will mimic: [latex]f(x)=x^2, \\quad f(x)=-x^2,\\quad f(x)=x^3,\\quad[\/latex] or [latex]\\quad f(x)=-x^3[\/latex].<\/p>\n<p>The same idea applies to <em>the ratio of leading terms\u00a0<\/em>of a\u00a0rational function.<\/p>\n<\/section>\n<p>There are three distinct outcomes when checking for horizontal asymptotes:<\/p>\n<p><strong>Case 1:<\/strong> If the degree of the denominator is greater than the degree of the numerator, there is a <strong>horizontal asymptote<\/strong> at [latex]y=0[\/latex].<\/p>\n<section class=\"textbox example\">\n<p>Consider the following function:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{4x+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\n<p>In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{4x}{{x}^{2}}=\\frac{4}{x}[\/latex].<\/p>\n<p>This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\\left(x\\right)=\\frac{4}{x}[\/latex], and the outputs will approach zero, resulting in a horizontal asymptote at [latex]y=0[\/latex]. Note that this graph crosses the horizontal asymptote.<\/p>\n<figure style=\"width: 900px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213933\/CNX_Precalc_Figure_03_07_0132.jpg\" alt=\"Graph of f(x)=(4x+2)\/(x^2+4x-5) with its vertical asymptotes at x=-5 and x=1 and its horizontal asymptote at y=0.\" width=\"900\" height=\"302\" \/><figcaption class=\"wp-caption-text\">Horizontal Asymptote [latex]y=0[\/latex] when [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne{0}\\text{ where degree of }p&lt;\\text{degree of q}[\/latex].<\/figcaption><\/figure>\n<\/section>\n<p><strong>Case 2:<\/strong> If the degree of the denominator is less than the degree of the numerator by one, we get a <strong>slant asymptote<\/strong>.<\/p>\n<section class=\"textbox example\">\n<p>Consider the following function:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{3{x}^{2}-2x+1}{x - 1}[\/latex]<\/p>\n<p>In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{x}=3x[\/latex].<\/p>\n<p>This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\\left(x\\right)=3x[\/latex]. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote.<\/p>\n<p>However, the graph of [latex]g\\left(x\\right)=3x[\/latex] looks like a diagonal line, and since [latex]f[\/latex] will behave similarly to [latex]g[\/latex], it will approach a line close to [latex]y=3x[\/latex]. This line is a slant asymptote.<\/p>\n<p>To find the equation of the slant asymptote, divide [latex]\\dfrac{3{x}^{2}-2x+1}{x - 1}[\/latex]. The quotient is [latex]3x+1[\/latex] and the remainder is [latex]2[\/latex]. The slant asymptote is the graph of the line [latex]g(x)=3x+1[\/latex].<\/p>\n<figure id=\"attachment_1823\" aria-describedby=\"caption-attachment-1823\" style=\"width: 532px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1823 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29160914\/Screenshot-2024-04-29-120903.png\" alt=\"Graph of f(x)=(3x^2-2x+1)\/(x-1) with its vertical asymptote at x=1 and a slant asymptote aty=3x+1.\" width=\"532\" height=\"539\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29160914\/Screenshot-2024-04-29-120903.png 532w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29160914\/Screenshot-2024-04-29-120903-296x300.png 296w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29160914\/Screenshot-2024-04-29-120903-65x66.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29160914\/Screenshot-2024-04-29-120903-225x228.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29160914\/Screenshot-2024-04-29-120903-350x355.png 350w\" sizes=\"(max-width: 532px) 100vw, 532px\" \/><figcaption id=\"caption-attachment-1823\" class=\"wp-caption-text\">Slant asymptote when [latex]f(x)=\\frac{p(x)}{q(x)}, q(x) \\neq 0[\/latex] where degree of [latex] p &gt; \\text{ degree of } q \\text{ by } 1[\/latex]<\/figcaption><\/figure>\n<\/section>\n<p><strong>Case 3:<\/strong> If the degree of the denominator is equal to the degree of the numerator, there is a horizontal asymptote at [latex]y=\\frac{{a}_{n}}{{b}_{n}}[\/latex], where [latex]{a}_{n}[\/latex] and [latex]{b}_{n}[\/latex] are the leading coefficients of [latex]p\\left(x\\right)[\/latex] and [latex]q\\left(x\\right)[\/latex] for [latex]f\\left(x\\right)=\\frac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0[\/latex].<\/p>\n<section class=\"textbox example\">\n<p>Consider the following function:<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{3{x}^{2}+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\n<p style=\"text-align: left;\">In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{{x}^{2}}=3[\/latex].<\/p>\n<p style=\"text-align: left;\">This tells us that as the inputs grow large, this function will behave like the function [latex]g\\left(x\\right)=3[\/latex], which is a horizontal line. As [latex]x\\to \\pm \\infty ,f\\left(x\\right)\\to 3[\/latex], resulting in a horizontal asymptote at [latex]y=3[\/latex]. Note that this graph crosses the horizontal asymptote.<\/p>\n<figure id=\"attachment_1826\" aria-describedby=\"caption-attachment-1826\" style=\"width: 802px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1826\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29161246\/Screenshot-2024-04-29-121237.png\" alt=\"Graph of f(x)=(3x^2+2)\/(x^2+4x-5) with its vertical asymptotes at x=-5 and x=1 and its horizontal asymptote at y=3.\" width=\"802\" height=\"374\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29161246\/Screenshot-2024-04-29-121237.png 802w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29161246\/Screenshot-2024-04-29-121237-300x140.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29161246\/Screenshot-2024-04-29-121237-768x358.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29161246\/Screenshot-2024-04-29-121237-65x30.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29161246\/Screenshot-2024-04-29-121237-225x105.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/29161246\/Screenshot-2024-04-29-121237-350x163.png 350w\" sizes=\"(max-width: 802px) 100vw, 802px\" \/><figcaption id=\"caption-attachment-1826\" class=\"wp-caption-text\">Horizontal Asymptote when [latex]f\\left(x\\right)=\\frac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0\\text{ where degree of }p=\\text{degree of }q[\/latex].<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<\/section>\n<p>Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.<\/p>\n<section class=\"textbox proTip\">\n<p>It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction.<\/p>\n<p>For instance, if we had the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{3{x}^{5}-{x}^{2}}{x+3}[\/latex]<\/p>\n<p>with end behavior<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)\\approx \\dfrac{3{x}^{5}}{x}=3{x}^{4}[\/latex],<\/p>\n<p>the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.<\/p>\n<p style=\"text-align: center;\">As [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>horizontal asymptote<\/strong><\/h3>\n<p>The <strong>horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.<\/p>\n<ul id=\"fs-id1165137722720\">\n<li><strong>Case 1:<\/strong> Degree of denominator [latex]>[\/latex] degree of\u00a0 the numerator: horizontal asymptote at [latex]y=0[\/latex]<\/li>\n<li><strong>Case 2<\/strong>: Degree of denominator [latex]<[\/latex] degree of numerator by one: no horizontal asymptote; slant asymptote.\n\n\n\n<ul>\n<li>If the degree of the numerator is greater than the degree of the denominator by\u00a0<em>more than one<\/em>, the end behavior of the function&#8217;s graph will mimic that of the graph of the reduced ratio of leading terms.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Case 3<\/strong>: Degree of numerator [latex]=[\/latex]\u00a0degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p>For the functions below, identify the horizontal or slant asymptote.<\/p>\n<ol>\n<li>[latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]<\/li>\n<li>[latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q968793\">Show Solution<\/button><\/p>\n<div id=\"q968793\" class=\"hidden-answer\" style=\"display: none\">\n<p>For these solutions, we will use [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)}, q\\left(x\\right)\\ne 0[\/latex].<\/p>\n<ol>\n<li>\n<p>The degree of [latex]p[\/latex] and the degree of [latex]q[\/latex] are both equal to [latex]3[\/latex], so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at [latex]y=\\frac{6}{2}[\/latex] or [latex]y=3[\/latex].<\/p>\n<\/li>\n<li>\n<p>The degree of [latex]p=2[\/latex] and degree of [latex]q=1[\/latex]. Since [latex]p>q[\/latex] by [latex]1[\/latex], there is a slant asymptote found at [latex]\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_4515\" aria-describedby=\"caption-attachment-4515\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4515 size-medium\" style=\"text-align: center;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/21031600\/CNX_CAT_Figure_05_01_011-300x71.jpg\" alt=\"Synthetic division of x^2-4x+1 by x+2, resulting in x-6 with a remainder of 13\" width=\"300\" height=\"71\" \/><figcaption id=\"caption-attachment-4515\" class=\"wp-caption-text\">Synthetic division<\/figcaption><\/figure>\n<\/div>\n<p>The quotient is [latex]x - 6[\/latex] and the remainder is [latex]13[\/latex]. There is a slant asymptote at [latex]y=-x - 6[\/latex].<\/p>\n<\/li>\n<li>\n<p>The degree of [latex]p=2\\text{ }<[\/latex] degree of [latex]q=3[\/latex], so there is a horizontal asymptote [latex]y=0[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288268\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288268&theme=lumen&iframe_resize_id=ohm288268&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the horizontal and vertical asymptotes of the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\dfrac{\\left(x - 2\\right)\\left(x+3\\right)}{\\left(x - 1\\right)\\left(x+2\\right)\\left(x - 5\\right)}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q228875\">Show Solution<\/button><\/p>\n<div id=\"q228875\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, note that this function has no common factors, so there are no potential removable discontinuities.<\/p>\n<p>The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at [latex]x=1,-2,\\text{and }5[\/latex], indicating vertical asymptotes at these values.<\/p>\n<p>The numerator has degree [latex]2[\/latex], while the denominator has degree [latex]3[\/latex]. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to 0[\/latex]. This function will have a horizontal asymptote at [latex]y=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm105058\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=105058&theme=lumen&iframe_resize_id=ohm105058&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":484,"module-header":"background_you_need","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\/1300"}],"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":38,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1300\/revisions"}],"predecessor-version":[{"id":4746,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1300\/revisions\/4746"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/484"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1300\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1300"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1300"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1300"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}