{"id":2015,"date":"2024-07-02T20:17:36","date_gmt":"2024-07-02T20:17:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2015"},"modified":"2025-08-15T13:51:56","modified_gmt":"2025-08-15T13:51:56","slug":"rational-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/rational-functions-learn-it-4\/","title":{"raw":"Rational Functions: Learn It 4","rendered":"Rational Functions: Learn It 4"},"content":{"raw":"

Horizontal Asymptotes and Intercepts<\/h2>\r\nWhile vertical asymptotes describe the behavior of a graph as the output<\/em> gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input<\/em> gets 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 leading terms of the numerator and denominator functions.\r\n\r\n
We have learned that knowing the leading coefficient and degree of a polynomial function is useful when predicting its end behavior.\r\n[latex]\\\\[\/latex]\r\nIf 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].\r\n[latex]\\\\[\/latex]\r\nThe same idea applies to the ratio of leading terms\u00a0<\/em>of a\u00a0rational function.<\/section>There are three distinct outcomes when checking for horizontal asymptotes:\r\n
    \r\n \t
  • Case 1:<\/strong> If the degree of the denominator [latex]>[\/latex] degree of the numerator, there is a horizontal asymptote<\/strong> at [latex]y=0[\/latex].\r\n
    \r\n

    Example: [latex]f\\left(x\\right)=\\dfrac{4x+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\r\nIn this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{4x}{{x}^{2}}=\\frac{4}{x}[\/latex]. 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]\"Graph 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<\\text{degree of q}[\/latex].[\/caption]<\/section><\/li>\r\n \t

  • Case 2:<\/strong> If the degree of the denominator [latex]<[\/latex] degree of the numerator by one, we get a slant asymptote.\r\n
    \r\n

    Example: [latex]f\\left(x\\right)=\\dfrac{3{x}^{2}-2x+1}{x - 1}[\/latex]<\/p>\r\nIn this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{x}=3x[\/latex]. 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. 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 (NOTE: the graph of the function itself is just a \"sketch\" within the parameters of the asymptotes).\r\n

    <\/div>\r\n[caption id=\"attachment_4477\" align=\"aligncenter\" width=\"489\"]\"Graph Slant Asymptote when [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0[\/latex] where degree of [latex]p>\\text{ degree of }q\\text{ by }1[\/latex].[\/caption]<\/section><\/li>\r\n \t
  • Case 3:<\/strong> If the degree of the denominator [latex]=[\/latex] 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].\r\n
    \r\n

    Example: [latex]f\\left(x\\right)=\\dfrac{3{x}^{2}+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\r\n

    In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{{x}^{2}}=3[\/latex]. 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_4478\" align=\"aligncenter\" width=\"731\"]\"Graph 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]<\/section><\/li>\r\n<\/ul>\r\n

    Notice that, while the graph of a rational function will never cross a vertical asymptote<\/strong>, 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.<\/section>
    \r\n

    horizontal asymptotes of rational functions<\/h3>\r\nThe horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.\r\n
      \r\n \t
    • Case 1:<\/strong> Degree of numerator is less than<\/em> degree of denominator: horizontal asymptote at\u00a0[latex]y=0[\/latex]<\/li>\r\n \t
    • Case 2<\/strong>: Degree of numerator is greater than degree of denominator by one<\/em>: no horizontal asymptote; slant asymptote.\r\n
        \r\n \t
      • If the degree of the numerator is greater than the degree of the denominator by\u00a0more 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
      • Case 3<\/strong>: Degree of numerator is equal to<\/em> degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\r\n<\/ul>\r\n<\/section>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<\/strong> of the graph will mimic the behavior of the reduced end behavior fraction.\r\n\r\n
        You'll need to graph the rational function [latex]f\\left(x\\right)\\ =\\ \\dfrac{\\left(4x^a+2\\right)}{\\left(x^b+2x-3\\right)}[\/latex] using an online graphing calculator. Adjust the [latex]a[\/latex] and the\u00a0[latex]b[\/latex] values to obtain graphs with the following horizontal asymptotes:\r\n
          \r\n \t
        1. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give\u00a0a graph with a slant asymptote.<\/li>\r\n \t
        2. Find values of [latex]a[\/latex] and [latex]b[\/latex]\u00a0that will give a graph with a horizontal asymptote at [latex]y = 0[\/latex]<\/li>\r\n \t
        3. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give a graph with a horizontal asymptote that is the ratio of the leading coefficients of [latex]f(x)[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"918195\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"918195\"]\r\n\r\nAnswers will vary.\r\n
            \r\n \t
          1. Find values of [latex]a[\/latex] and [latex]b[\/latex]\u00a0that will give\u00a0a graph with a slant asymptote. \u00a0For example [latex]b = 3, a = 4[\/latex]<\/li>\r\n \t
          2. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give a graph with a horizontal asymptote at [latex]y = 0[\/latex]. \u00a0For example, [latex]b = 4, a = 2[\/latex]<\/li>\r\n \t
          3. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give a graph with a horizontal asymptote that is the ratio of the leading coefficients of [latex]f(x)[\/latex].For example, [latex]b = 4, a = 4[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
            For the functions below, identify the horizontal or slant asymptote.\r\n
              \r\n \t
            1. [latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]<\/li>\r\n \t
            2. [latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]<\/li>\r\n \t
            3. [latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"968793\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"968793\"]\r\n\r\nFor 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].\r\n
                \r\n \t
              1. [latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]: The degree of [latex]p[\/latex] and the degree of [latex]q[\/latex] are both equal to 3, so we can find the horizontal asymptote by taking the ratio of the leading terms.\r\nThere is a horizontal asymptote at [latex]y=\\frac{6}{2}[\/latex] or [latex]y=3[\/latex].<\/li>\r\n \t
              2. [latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]: The degree of [latex]p=2[\/latex] and degree of [latex]q=1[\/latex]. Since [latex]p>q[\/latex] by 1, there is no horizontal asymptote.\r\nHowever, we have a slant asymptote found at [latex]\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex].\r\n\r\n[caption id=\"attachment_4515\" align=\"aligncenter\" width=\"300\"]\"Synthetic Synthetic division[\/caption]\r\n\r\nThe quotient is [latex]x - 6[\/latex] and the remainder is [latex]13[\/latex]. There is a slant asymptote at [latex]y=x - 6[\/latex].<\/li>\r\n \t
              3. [latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]: The degree of [latex]p=2\\text{ }<[\/latex] degree of [latex]q=3[\/latex], so there is a horizontal asymptote [latex]y=0[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
                Find the horizontal and vertical asymptotes of the function\r\n

                [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[reveal-answer q=\"228875\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"228875\"]\r\n\r\nFirst, note that this function has no common factors, so there are no potential removable discontinuities.\r\n\r\nThe 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.<\/strong>\r\n\r\nThe numerator has degree 2, while the denominator has degree 3. 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].<\/strong>\r\n\r\nBelow is the graph of the function to confirm our findings.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"Graph Graph of f(x) with asymptotes labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

                [ohm2_question hide_question_numbers=1]24655[\/ohm2_question]<\/section>
                [ohm2_question hide_question_numbers=1]24657[\/ohm2_question]<\/section>
                [ohm2_question hide_question_numbers=1]24656[\/ohm2_question]<\/section>\r\n

                Finding Intercepts of Rational Functions<\/h3>\r\nRecall that we may always find any horizontal intercepts of the graph of a function by setting the output equal to zero. And we may always find any vertical intercept by setting the input equal to zero.\r\n\r\nIn a rational function of the form [latex]r(x)=\\dfrac{P(x)}{Q(x)}[\/latex],\r\n
                  \r\n \t
                • Find the vertical intercept (the [latex]y[\/latex]-intercept) by evaluating [latex]r(0)[\/latex]. That is, replace all the input variables with [latex]0[\/latex] and calculate the result.<\/li>\r\n \t
                • Find the horizontal intercept(s) (the [latex]x[\/latex]-intercepts) by solving [latex]r(x)=0[\/latex]. Since the function is undefined where the denominator equals zero], set the numerator equal to zero to find the horizontal intercepts of the function.<\/li>\r\n \t
                • Note that the graph of a rational function may not possess a vertical- or horizontal-intercept.<\/li>\r\n<\/ul>\r\n
                  \r\n

                  intercepts of rational functions<\/h3>\r\nA rational function<\/strong> will have a [latex]y[\/latex]-intercept when the input is zero, if the function is defined at zero. A rational function will not have a [latex]y[\/latex]-intercept if the function is not defined at zero.\r\n[latex]\\\\[\/latex]\r\n\r\nLikewise, a rational function will have [latex]x[\/latex]-intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, [latex]x[\/latex]-intercepts can only occur when the numerator of the rational function is equal to zero.\r\n\r\n<\/section>
                  Find the intercepts of [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].[reveal-answer q=\"418596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418596\"]We can find the [latex]y[\/latex]-intercept by evaluating the function at zero\r\n

                  [latex]\\begin{align}f\\left(0\\right)&=\\dfrac{\\left(0 - 2\\right)\\left(0+3\\right)}{\\left(0 - 1\\right)\\left(0+2\\right)\\left(0 - 5\\right)} \\\\[1mm] &=\\frac{-6}{10} \\\\[1mm] &=-\\frac{3}{5} \\\\[1mm] &=-0.6 \\end{align}[\/latex]<\/p>\r\nThe [latex]x[\/latex]-intercepts will occur when the function is equal to zero. A rational function is equal to 0 when the numerator is 0, as long as the denominator is not also 0.\r\n

                  [latex]\\begin{align} 0&=\\frac{\\left(x - 2\\right)\\left(x+3\\right)}{\\left(x - 1\\right)\\left(x+2\\right)\\left(x - 5\\right)} \\\\[1mm] 0&=\\left(x - 2\\right)\\left(x+3\\right)\\end{align}[\/latex]<\/p>\r\n

                  [latex]x=2, -3[\/latex]<\/p>\r\nThe [latex]y[\/latex]-intercept is [latex]\\left(0,-0.6\\right)[\/latex], the [latex]x[\/latex]-intercepts are [latex]\\left(2,0\\right)[\/latex] and [latex]\\left(-3,0\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"Graph Graph of f(x) with asymptotes labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

                  [ohm2_question hide_question_numbers=1]24658[\/ohm2_question]<\/section>","rendered":"

                  Horizontal Asymptotes and Intercepts<\/h2>\n

                  While vertical asymptotes describe the behavior of a graph as the output<\/em> gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input<\/em> gets 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 leading terms of the numerator and denominator functions.<\/p>\n

                  We have learned that knowing the leading coefficient and degree of a polynomial function is useful when predicting its end behavior.
                  \n[latex]\\\\[\/latex]
                  \nIf 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].
                  \n[latex]\\\\[\/latex]
                  \nThe same idea applies to the ratio of leading terms\u00a0<\/em>of a\u00a0rational function.<\/section>\n

                  There are three distinct outcomes when checking for horizontal asymptotes:<\/p>\n

                    \n
                  • Case 1:<\/strong> If the degree of the denominator [latex]>[\/latex] degree of the numerator, there is a horizontal asymptote<\/strong> at [latex]y=0[\/latex].
                    \n
                    \n

                    Example: [latex]f\\left(x\\right)=\\dfrac{4x+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\n

                    In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{4x}{{x}^{2}}=\\frac{4}{x}[\/latex]. 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

                    \"Graph
                    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<\\text{degree of q}[\/latex].<\/figcaption><\/figure>\n<\/section>\n<\/li>\n
                  • Case 2:<\/strong> If the degree of the denominator [latex]<[\/latex] degree of the numerator by one, we get a slant asymptote.\n\n\n
                    \n

                    Example: [latex]f\\left(x\\right)=\\dfrac{3{x}^{2}-2x+1}{x - 1}[\/latex]<\/p>\n

                    In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{x}=3x[\/latex]. 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. 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 (NOTE: the graph of the function itself is just a “sketch” within the parameters of the asymptotes).<\/p>\n

                    <\/div>\n
                    \"Graph
                    Slant Asymptote when [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)},q\\left(x\\right)\\ne 0[\/latex] where degree of [latex]p>\\text{ degree of }q\\text{ by }1[\/latex].<\/figcaption><\/figure>\n<\/section>\n<\/li>\n
                  • Case 3:<\/strong> If the degree of the denominator [latex]=[\/latex] 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].
                    \n
                    \n

                    Example: [latex]f\\left(x\\right)=\\dfrac{3{x}^{2}+2}{{x}^{2}+4x - 5}[\/latex]<\/p>\n

                    In this case the end behavior is [latex]f\\left(x\\right)\\approx \\frac{3{x}^{2}}{{x}^{2}}=3[\/latex]. 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

                    \"Graph
                    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<\/section>\n<\/li>\n<\/ul>\n
                    Notice that, while the graph of a rational function will never cross a vertical asymptote<\/strong>, 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.<\/section>\n
                    \n

                    horizontal asymptotes of rational functions<\/h3>\n

                    The horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.<\/p>\n

                      \n
                    • Case 1:<\/strong> Degree of numerator is less than<\/em> degree of denominator: horizontal asymptote at\u00a0[latex]y=0[\/latex]<\/li>\n
                    • Case 2<\/strong>: Degree of numerator is greater than degree of denominator by one<\/em>: no horizontal asymptote; slant asymptote.\n
                        \n
                      • If the degree of the numerator is greater than the degree of the denominator by\u00a0more 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>\n<\/ul>\n<\/li>\n
                      • Case 3<\/strong>: Degree of numerator is equal to<\/em> degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\n<\/ul>\n<\/section>\n

                        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<\/strong> of the graph will mimic the behavior of the reduced end behavior fraction.<\/p>\n

                        You’ll need to graph the rational function [latex]f\\left(x\\right)\\ =\\ \\dfrac{\\left(4x^a+2\\right)}{\\left(x^b+2x-3\\right)}[\/latex] using an online graphing calculator. Adjust the [latex]a[\/latex] and the\u00a0[latex]b[\/latex] values to obtain graphs with the following horizontal asymptotes:<\/p>\n
                          \n
                        1. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give\u00a0a graph with a slant asymptote.<\/li>\n
                        2. Find values of [latex]a[\/latex] and [latex]b[\/latex]\u00a0that will give a graph with a horizontal asymptote at [latex]y = 0[\/latex]<\/li>\n
                        3. Find values of [latex]a[\/latex] and [latex]b[\/latex] that will give a graph with a horizontal asymptote that is the ratio of the leading coefficients of [latex]f(x)[\/latex].<\/li>\n<\/ol>\n