{"id":2015,"date":"2018-01-11T21:13:20","date_gmt":"2018-01-11T21:13:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/lhopitals-rule\/"},"modified":"2019-02-22T19:54:53","modified_gmt":"2019-02-22T19:54:53","slug":"lhopitals-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/lhopitals-rule\/","title":{"raw":"4.8 L\u2019H\u00f4pital\u2019s Rule","rendered":"4.8 L\u2019H\u00f4pital\u2019s Rule"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Recognize when to apply L\u2019H\u00f4pital\u2019s rule.<\/li>\r\n \t<li>Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L\u2019H\u00f4pital\u2019s rule in each case.<\/li>\r\n \t<li>Describe the relative growth rates of functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165042709572\">In this section, we examine a powerful tool for evaluating limits. This tool, known as <strong>L\u2019H\u00f4pital\u2019s rule<\/strong>, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.<\/p>\r\n\r\n<div id=\"fs-id1165043085155\" class=\"bc-section section\">\r\n<h1>Applying L\u2019H\u00f4pital\u2019s Rule<\/h1>\r\n<p id=\"fs-id1165042941863\">L\u2019H\u00f4pital\u2019s rule can be used to evaluate limits involving the quotient of two functions. Consider<\/p>\r\n\r\n<div id=\"fs-id1165042458008\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}[\/latex].<\/div>\r\n<p id=\"fs-id1165042613148\">If [latex]\\underset{x\\to a}{\\lim}f(x)=L_1[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=L_2 \\ne 0[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1165043088102\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\frac{L_1}{L_2}[\/latex].<\/div>\r\n<p id=\"fs-id1165042330308\">However, what happens if [latex]\\underset{x\\to a}{\\lim}f(x)=0[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=0[\/latex]? We call this one of the <strong>indeterminate forms<\/strong>, of type [latex]\\frac{0}{0}[\/latex]. This is considered an indeterminate form because we cannot determine the exact behavior of [latex]\\frac{f(x)}{g(x)}[\/latex] as [latex]x\\to a[\/latex] without further analysis. We have seen examples of this earlier in the text. For example, consider<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to 2}{\\lim}\\frac{x^2-4}{x-2}[\/latex] and [latex]\\underset{x\\to 0}{\\lim}\\frac{ \\sin x}{x}[\/latex].<\/div>\r\n<p id=\"fs-id1165043036412\">For the first of these examples, we can evaluate the limit by factoring the numerator and writing<\/p>\r\n\r\n<div id=\"fs-id1165043067705\" class=\"equation unnumbered\">[latex]\\underset{x\\to 2}{\\lim}\\frac{x^2-4}{x-2}=\\underset{x\\to 2}{\\lim}\\frac{(x+2)(x-2)}{x-2}=\\underset{x\\to 2}{\\lim}(x+2)=2+2=4[\/latex].<\/div>\r\nFor [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}[\/latex] we were able to show, using a geometric argument, that\r\n<div id=\"fs-id1165042954700\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}=1[\/latex].<\/div>\r\n<p id=\"fs-id1165043001194\">Here we use a different technique for evaluating limits such as these. Not only does this technique provide an easier way to evaluate these limits, but also, and more important, it provides us with a way to evaluate many other limits that we could not calculate previously.<\/p>\r\n<p id=\"fs-id1165043062373\">The idea behind L\u2019H\u00f4pital\u2019s rule can be explained using local linear approximations. Consider two differentiable functions [latex]f[\/latex] and [latex]g[\/latex] such that [latex]\\underset{x\\to a}{\\lim}f(x)=0=\\underset{x\\to a}{\\lim}g(x)[\/latex] and such that [latex]g^{\\prime}(a)\\ne 0[\/latex] For [latex]x[\/latex] near [latex]a[\/latex], we can write<\/p>\r\n\r\n<div id=\"fs-id1165043199940\" class=\"equation unnumbered\">[latex]f(x)\\approx f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/div>\r\n<p id=\"fs-id1165042355300\">and<\/p>\r\n\r\n<div id=\"fs-id1165042320393\" class=\"equation unnumbered\">[latex]g(x)\\approx g(a)+g^{\\prime}(a)(x-a)[\/latex].<\/div>\r\n<p id=\"fs-id1165043178271\">Therefore,<\/p>\r\n\r\n<div id=\"fs-id1165042331440\" class=\"equation unnumbered\">[latex]\\frac{f(x)}{g(x)}\\approx \\frac{f(a)+f^{\\prime}(a)(x-a)}{g(a)+g^{\\prime}(a)(x-a)}[\/latex].<\/div>\r\n<div id=\"CNX_Calc_Figure_04_08_001\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"618\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211304\/CNX_Calc_Figure_04_08_003.jpg\" alt=\"Two functions y = f(x) and y = g(x) are drawn such that they cross at a point above x = a. The linear approximations of these two functions y = f(a) + f\u2019(a)(x \u2013 a) and y = g(a) + g\u2019(a)(x \u2013 a) are also drawn.\" width=\"618\" height=\"390\" \/> <strong>Figure 1.<\/strong> If [latex]\\underset{x\\to a}{\\lim}f(x)=\\underset{x\\to a}{\\lim}g(x)[\/latex], then the ratio [latex]f(x)\/g(x)[\/latex] is approximately equal to the ratio of their linear approximations near [latex]a[\/latex].[\/caption]<\/div>\r\nSince [latex]f[\/latex] is differentiable at [latex]a[\/latex], then [latex]f[\/latex] is continuous at [latex]a[\/latex], and therefore [latex]f(a)=\\underset{x\\to a}{\\lim}f(x)=0[\/latex]. Similarly, [latex]g(a)=\\underset{x\\to a}{\\lim}g(x)=0[\/latex]. If we also assume that [latex]f^{\\prime}[\/latex] and [latex]g^{\\prime}[\/latex] are continuous at [latex]x=a[\/latex], then [latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}f^{\\prime}(x)[\/latex] and [latex]g^{\\prime}(a)=\\underset{x\\to a}{\\lim}g^{\\prime}(x)[\/latex]. Using these ideas, we conclude that\r\n<div id=\"fs-id1165042373953\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)(x-a)}{g^{\\prime}(x)(x-a)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex].<\/div>\r\nNote that the assumption that [latex]f^{\\prime}[\/latex] and [latex]g^{\\prime}[\/latex] are continuous at [latex]a[\/latex] and [latex]g^{\\prime}(a)\\ne 0[\/latex] can be loosened. We state L\u2019H\u00f4pital\u2019s rule formally for the indeterminate form [latex]\\frac{0}{0}[\/latex]. Also note that the notation [latex]\\frac{0}{0}[\/latex] does not mean we are actually dividing zero by zero. Rather, we are using the notation [latex]\\frac{0}{0}[\/latex] to represent a quotient of limits, each of which is zero.\r\n<div id=\"fs-id1165043352593\" class=\"textbox key-takeaways theorem\">\r\n<h3>L\u2019H\u00f4pital\u2019s Rule (0\/0 Case)<\/h3>\r\n<p id=\"fs-id1165043276140\">Suppose [latex]f[\/latex] and [latex]g[\/latex] are differentiable functions over an open interval containing [latex]a[\/latex], except possibly at [latex]a[\/latex]. If [latex]\\underset{x\\to a}{\\lim}f(x)=0[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=0[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1165043020148\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex],<\/div>\r\n<p id=\"fs-id1165043035673\">assuming the limit on the right exists or is [latex]\\infty [\/latex] or [latex]\u2212\\infty[\/latex]. This result also holds if we are considering one-sided limits, or if [latex]a=\\infty[\/latex] or [latex]-\\infty[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042712928\" class=\"bc-section section\">\r\n<h2>Proof<\/h2>\r\n<p id=\"fs-id1165043429940\">We provide a proof of this theorem in the special case when [latex]f, \\, g, \\, f^{\\prime}[\/latex], and [latex]g^{\\prime}[\/latex] are all continuous over an open interval containing [latex]a[\/latex]. In that case, since [latex]\\underset{x\\to a}{\\lim}f(x)=0=\\underset{x\\to a}{\\lim}g(x)[\/latex] and [latex]f[\/latex] and [latex]g[\/latex] are continuous at [latex]a[\/latex], it follows that [latex]f(a)=0=g(a)[\/latex]. Therefore,<\/p>\r\n\r\n<div id=\"fs-id1165043353664\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll} \\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)} &amp; =\\underset{x\\to a}{\\lim}\\frac{f(x)-f(a)}{g(x)-g(a)} &amp; &amp; &amp; \\text{since} \\, f(a)=0=g(a) \\\\ &amp; =\\underset{x\\to a}{\\lim}\\frac{\\frac{f(x)-f(a)}{x-a}}{\\frac{g(x)-g(a)}{x-a}} &amp; &amp; &amp; \\text{multiplying numerator and denominator by} \\, \\frac{1}{x-a} \\\\ &amp; =\\frac{\\underset{x\\to a}{\\lim}\\frac{f(x)-f(a)}{x-a}}{\\underset{x\\to a}{\\lim}\\frac{g(x)-g(a)}{x-a}} &amp; &amp; &amp; \\text{limit of a quotient} \\\\ &amp; =\\frac{f^{\\prime}(a)}{g^{\\prime}(a)} &amp; &amp; &amp; \\text{definition of the derivative} \\\\ &amp; =\\frac{\\underset{x\\to a}{\\lim}f^{\\prime}(x)}{\\underset{x\\to a}{\\lim}g^{\\prime}(x)} &amp; &amp; &amp; \\text{continuity of} \\, f^{\\prime} \\, \\text{and} \\, g^{\\prime} \\\\ &amp; =\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)} &amp; &amp; &amp; \\text{limit of a quotient} \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165043092431\">Note that L\u2019H\u00f4pital\u2019s rule states we can calculate the limit of a quotient [latex]\\frac{f}{g}[\/latex] by considering the limit of the quotient of the derivatives [latex]\\frac{f^{\\prime}}{g^{\\prime}}[\/latex]. It is important to realize that we are not calculating the derivative of the quotient [latex]\\frac{f}{g}[\/latex].<\/p>\r\n<p id=\"fs-id1165042988256\">\u25a1<\/p>\r\n\r\n<div id=\"fs-id1165043397578\" class=\"textbox examples\">\r\n<h3>Applying L\u2019H\u00f4pital\u2019s Rule (0\/0 Case)<\/h3>\r\n<div id=\"fs-id1165043104016\" class=\"exercise\">\r\n<div id=\"fs-id1165043395456\" class=\"textbox\">\r\n<p id=\"fs-id1165042456913\">Evaluate each of the following limits by applying L\u2019H\u00f4pital\u2019s rule.<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{1- \\cos x}{x}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 1}{\\lim}\\frac{\\sin (\\pi x)}{\\ln x}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^{1\/x}-1}{1\/x}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042535045\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042535045\"]\r\n<ol id=\"fs-id1165042535045\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Since the numerator [latex]1- \\cos x\\to 0[\/latex] and the denominator [latex]x\\to 0[\/latex], we can apply L\u2019H\u00f4pital\u2019s rule to evaluate this limit. We have\r\n<div id=\"fs-id1165042328696\" class=\"equation unnumbered\">[latex]\\begin{array}{ll} \\underset{x\\to 0}{\\lim}\\frac{1- \\cos x}{x} &amp; =\\underset{x\\to 0}{\\lim}\\frac{\\frac{d}{dx}(1- \\cos x)}{\\frac{d}{dx}(x)} \\\\ &amp; =\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{1} \\\\ &amp; =\\frac{\\underset{x\\to 0}{\\lim}(\\sin x)}{\\underset{x\\to 0}{\\lim}(1)} \\\\ &amp; =\\frac{0}{1}=0 \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>As [latex]x\\to 1[\/latex], the numerator [latex]\\sin (\\pi x)\\to 0[\/latex] and the denominator [latex]\\ln x \\to 0[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule. We obtain\r\n<div id=\"fs-id1165043116372\" class=\"equation unnumbered\">[latex]\\begin{array}{ll} \\underset{x\\to 1}{\\lim}\\frac{\\sin (\\pi x)}{\\ln x} &amp; =\\underset{x\\to 1}{\\lim}\\frac{\\pi \\cos (\\pi x)}{1\/x} \\\\ &amp; =\\underset{x\\to 1}{\\lim}(\\pi x) \\cos (\\pi x) \\\\ &amp; =(\\pi \\cdot 1)(-1)=\u2212\\pi \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>As [latex]x\\to \\infty[\/latex], the numerator [latex]e^{1\/x}-1\\to 0[\/latex] and the denominator [latex](\\frac{1}{x})\\to 0[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule. We obtain\r\n<div id=\"fs-id1165042327460\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^{1\/x}-1}{\\frac{1}{x}}=\\underset{x\\to \\infty }{\\lim}\\frac{e^{1\/x}(\\frac{-1}{x^2})}{(\\frac{-1}{x^2})}=\\underset{x\\to \\infty}{\\lim} e^{1\/x}=e^0=1[\/latex]<\/div><\/li>\r\n \t<li>As [latex]x\\to 0[\/latex], both the numerator and denominator approach zero. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule. We obtain\r\n<div id=\"fs-id1165042562974\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}=\\underset{x\\to 0}{\\lim}\\frac{\\cos x-1}{2x}[\/latex].<\/div>\r\nSince the numerator and denominator of this new quotient both approach zero as [latex]x\\to 0[\/latex], we apply L\u2019H\u00f4pital\u2019s rule again. In doing so, we see that\r\n<div id=\"fs-id1165043281524\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\cos x-1}{2x}=\\underset{x\\to 0}{\\lim}\\frac{\u2212\\sin x}{2}=0[\/latex].<\/div>\r\nTherefore, we conclude that\r\n<div id=\"fs-id1165043284142\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}=0[\/latex].<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043393593\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165043395209\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1165043395214\">Evaluate [latex]\\underset{x\\to 0}{\\lim}\\frac{x}{\\tan x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042377480\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042377480\"]\r\n<p id=\"fs-id1165042377480\">1<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043395671\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043395678\">[latex]\\frac{d}{dx} \\tan x= \\sec ^2 x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042318672\">We can also use L\u2019H\u00f4pital\u2019s rule to evaluate limits of quotients [latex]\\frac{f(x)}{g(x)}[\/latex] in which [latex]f(x)\\to \\pm \\infty [\/latex] and [latex]g(x)\\to \\pm \\infty[\/latex]. Limits of this form are classified as <em>indeterminate forms of type<\/em> [latex]\\infty \/ \\infty[\/latex]. Again, note that we are not actually dividing [latex]\\infty[\/latex] by [latex]\\infty[\/latex]. Since [latex]\\infty[\/latex] is not a real number, that is impossible; rather, [latex]\\infty \/ \\infty[\/latex] is used to represent a quotient of limits, each of which is [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165042330826\" class=\"textbox key-takeaways theorem\">\r\n<h3>L\u2019H\u00f4pital\u2019s Rule ([latex]\\infty \/ \\infty[\/latex] Case)<\/h3>\r\n<p id=\"fs-id1165043426174\">Suppose [latex]f[\/latex] and [latex]g[\/latex] are differentiable functions over an open interval containing [latex]a[\/latex], except possibly at [latex]a[\/latex]. Suppose [latex]\\underset{x\\to a}{\\lim}f(x)=\\infty[\/latex] (or [latex]\u2212\\infty[\/latex]) and [latex]\\underset{x\\to a}{\\lim}g(x)=\\infty[\/latex] (or [latex]\u2212\\infty[\/latex]). Then,<\/p>\r\n\r\n<div id=\"fs-id1165042373703\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex],<\/div>\r\n<p id=\"fs-id1165042707280\">assuming the limit on the right exists or is [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex]. This result also holds if the limit is infinite, if [latex]a=\\infty[\/latex] or [latex]\u2212\\infty[\/latex], or the limit is one-sided.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043427623\" class=\"textbox examples\">\r\n<h3>Applying L\u2019H\u00f4pital\u2019s Rule ([latex]\\infty \/\\infty[\/latex] Case)<\/h3>\r\n<div id=\"fs-id1165043427625\" class=\"exercise\">\r\n<div id=\"fs-id1165042323490\" class=\"textbox\">\r\n<p id=\"fs-id1165042709794\">Evaluate each of the following limits by applying L\u2019H\u00f4pital\u2019s rule.<\/p>\r\n\r\n<ol id=\"fs-id1165042709798\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\underset{x\\to \\infty }{\\lim}\\frac{3x+5}{2x+1}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{\\cot x}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042376758\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042376758\"]\r\n<ol id=\"fs-id1165042376758\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Since [latex]3x+5[\/latex] and [latex]2x+1[\/latex] are first-degree polynomials with positive leading coefficients, [latex]\\underset{x\\to \\infty }{\\lim}(3x+5)=\\infty [\/latex] and [latex]\\underset{x\\to \\infty }{\\lim}(2x+1)=\\infty[\/latex]. Therefore, we apply L\u2019H\u00f4pital\u2019s rule and obtain\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty}{\\lim}\\frac{3x+5}{2x+1}=\\underset{x\\to \\infty }{\\lim}\\frac{3}{2}=\\frac{3}{2}[\/latex].<\/div>\r\nNote that this limit can also be calculated without invoking L\u2019H\u00f4pital\u2019s rule. Earlier in the chapter we showed how to evaluate such a limit by dividing the numerator and denominator by the highest power of [latex]x[\/latex] in the denominator. In doing so, we saw that\r\n<div id=\"fs-id1165042319986\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{3x+5}{2x+1}=\\underset{x\\to \\infty }{\\lim}\\frac{3+5\/x}{2+1\/x}=\\frac{3}{2}[\/latex].<\/div>\r\nL\u2019H\u00f4pital\u2019s rule provides us with an alternative means of evaluating this type of limit.<\/li>\r\n \t<li>Here, [latex]\\underset{x\\to 0^+}{\\lim} \\ln x=\u2212\\infty [\/latex] and [latex]\\underset{x\\to 0^+}{\\lim} \\cot x=\\infty[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule and obtain\r\n<div id=\"fs-id1165042374803\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{\\cot x}=\\underset{x\\to 0^+}{\\lim}\\frac{1\/x}{\u2212\\csc^2 x}=\\underset{x\\to 0^+}{\\lim}\\frac{1}{\u2212x \\csc^2 x}[\/latex].<\/div>\r\nNow as [latex]x\\to 0^+[\/latex], [latex]\\csc^2 x\\to \\infty[\/latex]. Therefore, the first term in the denominator is approaching zero and the second term is getting really large. In such a case, anything can happen with the product. Therefore, we cannot make any conclusion yet. To evaluate the limit, we use the definition of [latex]\\csc x[\/latex] to write\r\n<div id=\"fs-id1165042376466\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{\u2212x \\csc^2 x}=\\underset{x\\to 0^+}{\\lim}\\frac{\\sin^2 x}{\u2212x}[\/latex].<\/div>\r\nNow [latex]\\underset{x\\to 0^+}{\\lim} \\sin^2 x=0[\/latex] and [latex]\\underset{x\\to 0^+}{\\lim} x=0[\/latex], so we apply L\u2019H\u00f4pital\u2019s rule again. We find\r\n<div id=\"fs-id1165043249678\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\sin^2 x}{\u2212x}=\\underset{x\\to 0^+}{\\lim}\\frac{2 \\sin x \\cos x}{-1}=\\frac{0}{-1}=0[\/latex].<\/div>\r\nWe conclude that\r\n<div id=\"fs-id1165042315721\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{\\cot x}=0[\/latex].<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042333392\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042708180\" class=\"exercise\">\r\n<div id=\"fs-id1165042708182\" class=\"textbox\">\r\n<p id=\"fs-id1165042708184\">Evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{5x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042367881\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042367881\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042367881\"]0<\/div>\r\n<div id=\"fs-id1165042367890\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042705514\">[latex]\\frac{d}{dx}\\ln x=\\frac{1}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042320327\">As mentioned, L\u2019H\u00f4pital\u2019s rule is an extremely useful tool for evaluating limits. It is important to remember, however, that to apply L\u2019H\u00f4pital\u2019s rule to a quotient [latex]\\frac{f(x)}{g(x)}[\/latex], it is essential that the limit of [latex]\\frac{f(x)}{g(x)}[\/latex] be of the form [latex]0\/0[\/latex] or [latex]\\infty \/ \\infty[\/latex] Consider the following example.<\/p>\r\n\r\n<div id=\"fs-id1165042350228\" class=\"textbox examples\">\r\n<h3>When L\u2019H\u00f4pital\u2019s Rule Does Not Apply<\/h3>\r\n<div id=\"fs-id1165043219076\" class=\"exercise\">\r\n<div id=\"fs-id1165043219079\" class=\"textbox\">\r\n<p id=\"fs-id1165043219084\">Consider [latex]\\underset{x\\to 1}{\\lim}\\frac{x^2+5}{3x+4}[\/latex]. Show that the limit cannot be evaluated by applying L\u2019H\u00f4pital\u2019s rule.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042327372\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042327372\"]\r\n<p id=\"fs-id1165042327372\">Because the limits of the numerator and denominator are not both zero and are not both infinite, we cannot apply L\u2019H\u00f4pital\u2019s rule. If we try to do so, we get<\/p>\r\n\r\n<div id=\"fs-id1165042398961\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(x^2+5)=2x[\/latex]<\/div>\r\n<p id=\"fs-id1165043395079\">and<\/p>\r\n\r\n<div id=\"fs-id1165042318693\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(3x+4)=3[\/latex].<\/div>\r\n<p id=\"fs-id1165042364614\">At which point we would conclude erroneously that<\/p>\r\n\r\n<div id=\"fs-id1165042364618\" class=\"equation unnumbered\">[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2+5}{3x+4}=\\underset{x\\to 1}{\\lim}\\frac{2x}{3}=\\frac{2}{3}[\/latex].<\/div>\r\n<p id=\"fs-id1165043222028\">However, since [latex]\\underset{x\\to 1}{\\lim}(x^2+5)=6[\/latex] and [latex]\\underset{x\\to 1}{\\lim}(3x+4)=7[\/latex], we actually have<\/p>\r\n\r\n<div id=\"fs-id1165042970462\" class=\"equation unnumbered\">[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2+5}{3x+4}=\\frac{6}{7}[\/latex].<\/div>\r\n<p id=\"fs-id1165042383150\">We can conclude that<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2+5}{3x+4}\\ne \\underset{x\\to 1}{\\lim}\\frac{\\frac{d}{dx}(x^2+5)}{\\frac{d}{dx}(3x+4)}[\/latex].[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043174646\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165043174649\" class=\"exercise\">\r\n<div id=\"fs-id1165043174651\" class=\"textbox\">\r\n<p id=\"fs-id1165043174654\">Explain why we cannot apply L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\cos x}{x}[\/latex]. Evaluate [latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\cos x}{x}[\/latex] by other means.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042632518\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042632518\"]\r\n<p id=\"fs-id1165042632518\">[latex]\\underset{x\\to 0^+}{\\lim} \\cos x=1[\/latex]. Therefore, we cannot apply L\u2019H\u00f4pital\u2019s rule. The limit of the quotient is [latex]\\infty [\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042632611\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042632618\">Determine the limits of the numerator and denominator separately.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042331793\" class=\"bc-section section\">\r\n<h1>Other Indeterminate Forms<\/h1>\r\n<p id=\"fs-id1165042331798\">L\u2019H\u00f4pital\u2019s rule is very useful for evaluating limits involving the indeterminate forms [latex]0\/0[\/latex] and [latex]\\infty \/ \\infty[\/latex]. However, we can also use L\u2019H\u00f4pital\u2019s rule to help evaluate limits involving other indeterminate forms that arise when evaluating limits. The expressions [latex]0 \\cdot \\infty[\/latex], [latex]\\infty - \\infty[\/latex], [latex]1^{\\infty}[\/latex], [latex]\\infty^0[\/latex], and [latex]0^0[\/latex] are all considered indeterminate forms. These expressions are not real numbers. Rather, they represent forms that arise when trying to evaluate certain limits. Next we realize why these are indeterminate forms and then understand how to use L\u2019H\u00f4pital\u2019s rule in these cases. The key idea is that we must rewrite the indeterminate forms in such a way that we arrive at the indeterminate form [latex]0\/0[\/latex] or [latex]\\infty \/ \\infty[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165042320288\" class=\"bc-section section\">\r\n<h2>Indeterminate Form of Type [latex]0 \\cdot \\infty[\/latex]<\/h2>\r\n<p id=\"fs-id1165042320301\">Suppose we want to evaluate [latex]\\underset{x\\to a}{\\lim}(f(x) \\cdot g(x))[\/latex], where [latex]f(x)\\to 0[\/latex] and [latex]g(x)\\to \\infty[\/latex] (or [latex]\u2212\\infty[\/latex]) as [latex]x\\to a[\/latex]. Since one term in the product is approaching zero but the other term is becoming arbitrarily large (in magnitude), anything can happen to the product. We use the notation [latex]0 \\cdot \\infty[\/latex] to denote the form that arises in this situation. The expression [latex]0 \\cdot \\infty[\/latex] is considered indeterminate because we cannot determine without further analysis the exact behavior of the product [latex]f(x)g(x)[\/latex] as [latex]x\\to \\infty[\/latex]. For example, let [latex]n[\/latex] be a positive integer and consider<\/p>\r\n\r\n<div id=\"fs-id1165043259749\" class=\"equation unnumbered\">[latex]f(x)=\\frac{1}{(x^n+1)}[\/latex] and [latex]g(x)=3x^2[\/latex].<\/div>\r\n<p id=\"fs-id1165042323522\">As [latex]x\\to \\infty[\/latex], [latex]f(x)\\to 0[\/latex] and [latex]g(x)\\to \\infty[\/latex]. However, the limit as [latex]x\\to \\infty [\/latex] of [latex]f(x)g(x)=\\frac{3x^2}{(x^n+1)}[\/latex] varies, depending on [latex]n[\/latex]. If [latex]n=2[\/latex], then [latex]\\underset{x\\to \\infty }{\\lim}f(x)g(x)=3[\/latex]. If [latex]n=1[\/latex], then [latex]\\underset{x\\to \\infty }{\\lim}f(x)g(x)=\\infty[\/latex]. If [latex]n=3[\/latex], then [latex]\\underset{x\\to \\infty }{\\lim}f(x)g(x)=0[\/latex]. Here we consider another limit involving the indeterminate form [latex]0 \\cdot \\infty[\/latex] and show how to rewrite the function as a quotient to use L\u2019H\u00f4pital\u2019s rule.<\/p>\r\n\r\n<div id=\"fs-id1165042368495\" class=\"textbox examples\">\r\n<h3>Indeterminate Form of Type [latex]0\u00b7\\infty [\/latex]<\/h3>\r\n<div id=\"fs-id1165042368497\" class=\"exercise\">\r\n<div id=\"fs-id1165042323649\" class=\"textbox\">\r\n<p id=\"fs-id1165042323663\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim}x \\ln x[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042545829\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042545829\"]\r\n<p id=\"fs-id1165042545829\">First, rewrite the function [latex]x \\ln x[\/latex] as a quotient to apply L\u2019H\u00f4pital\u2019s rule. If we write<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]x \\ln x=\\frac{\\ln x}{1\/x}[\/latex],<\/div>\r\n<p id=\"fs-id1165042383922\">we see that [latex]\\ln x\\to \u2212\\infty [\/latex] as [latex]x\\to 0^+[\/latex] and [latex]\\frac{1}{x}\\to \\infty [\/latex] as [latex]x\\to 0^+[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule and obtain<\/p>\r\n\r\n<div id=\"fs-id1165042705715\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{1\/x}=\\underset{x\\to 0^+}{\\lim}\\frac{\\frac{d}{dx}(\\ln x)}{\\frac{d}{dx}(1\/x)}=\\underset{x\\to 0^+}{\\lim}\\frac{1\/x}{-1\/x^2}=\\underset{x\\to 0^+}{\\lim}(\u2212x)=0[\/latex].<\/div>\r\n<p id=\"fs-id1165042318647\">We conclude that<\/p>\r\n\r\n<div id=\"fs-id1165042318650\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}x \\ln x=0[\/latex].<\/div>\r\n<div id=\"CNX_Calc_Figure_04_08_002\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"358\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211307\/CNX_Calc_Figure_04_08_004.jpg\" alt=\"The function y = x ln(x) is graphed for values x \u2265 0. At x = 0, the value of the function is 0.\" width=\"358\" height=\"347\" \/> <strong>Figure 2.<\/strong> Finding the limit at [latex]x=0[\/latex] of the function [latex]f(x)=x \\ln x[\/latex].[\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043430905\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165043430908\" class=\"exercise\">\r\n<div id=\"fs-id1165043430910\" class=\"textbox\">\r\n<p id=\"fs-id1165043430912\">Evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cot x[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043286669\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043286669\"]\r\n<p id=\"fs-id1165043286669\">1<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043286676\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043286682\">Write [latex]x \\cot x=\\frac{x \\cos x}{\\sin x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042318802\" class=\"bc-section section\">\r\n<h2>Indeterminate Form of Type [latex]\\infty -\\infty[\/latex]<\/h2>\r\n<p id=\"fs-id1165042318816\">Another type of indeterminate form is [latex]\\infty -\\infty[\/latex]. Consider the following example. Let [latex]n[\/latex] be a positive integer and let [latex]f(x)=3x^n[\/latex] and [latex]g(x)=3x^2+5[\/latex]. As [latex]x\\to \\infty[\/latex], [latex]f(x)\\to \\infty [\/latex] and [latex]g(x)\\to \\infty [\/latex]. We are interested in [latex]\\underset{x\\to \\infty}{\\lim}(f(x)-g(x))[\/latex]. Depending on whether [latex]f(x)[\/latex] grows faster, [latex]g(x)[\/latex] grows faster, or they grow at the same rate, as we see next, anything can happen in this limit. Since [latex]f(x)\\to \\infty [\/latex] and [latex]g(x)\\to \\infty[\/latex], we write [latex]\\infty -\\infty [\/latex] to denote the form of this limit. As with our other indeterminate forms, [latex]\\infty -\\infty [\/latex] has no meaning on its own and we must do more analysis to determine the value of the limit. For example, suppose the exponent [latex]n[\/latex] in the function [latex]f(x)=3x^n[\/latex] is [latex]n=3[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1165043430807\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}(f(x)-g(x))=\\underset{x\\to \\infty }{\\lim}(3x^3-3x^2-5)=\\infty[\/latex].<\/div>\r\n<p id=\"fs-id1165042333220\">On the other hand, if [latex]n=2[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1165042333235\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}(f(x)-g(x))=\\underset{x\\to \\infty }{\\lim}(3x^2-3x^2-5)=-5[\/latex].<\/div>\r\n<p id=\"fs-id1165043395183\">However, if [latex]n=1[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1165043254251\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}(f(x)-g(x))=\\underset{x\\to \\infty }{\\lim}(3x-3x^2-5)=\u2212\\infty[\/latex].<\/div>\r\n<p id=\"fs-id1165043323851\">Therefore, the limit cannot be determined by considering only [latex]\\infty -\\infty[\/latex]. Next we see how to rewrite an expression involving the indeterminate form [latex]\\infty -\\infty [\/latex] as a fraction to apply L\u2019H\u00f4pital\u2019s rule.<\/p>\r\n\r\n<div id=\"fs-id1165043323875\" class=\"textbox examples\">\r\n<h3>Indeterminate Form of Type [latex]\\infty -\\infty[\/latex]<\/h3>\r\n<div id=\"fs-id1165043323877\" class=\"exercise\">\r\n<div id=\"fs-id1165043323879\" class=\"textbox\">\r\n<p id=\"fs-id1165042375663\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim}(\\frac{1}{x^2}-\\frac{1}{\\tan x})[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043281296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043281296\"]\r\n<p id=\"fs-id1165043281296\">By combining the fractions, we can write the function as a quotient. Since the least common denominator is [latex]x^2 \\tan x[\/latex], we have<\/p>\r\n\r\n<div id=\"fs-id1165043281316\" class=\"equation unnumbered\">[latex]\\frac{1}{x^2}-\\frac{1}{\\tan x}=\\frac{(\\tan x)-x^2}{x^2 \\tan x}[\/latex].<\/div>\r\n<p id=\"fs-id1165043259808\">As [latex]x\\to 0^+[\/latex], the numerator [latex]\\tan x-x^2 \\to 0[\/latex] and the denominator [latex]x^2 \\tan x \\to 0[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule. Taking the derivatives of the numerator and the denominator, we have<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{(\\tan x)-x^2}{x^2 \\tan x}=\\underset{x\\to 0^+}{\\lim}\\frac{(\\sec^2 x)-2x}{x^2 \\sec^2 x+2x \\tan x}[\/latex].<\/div>\r\n<p id=\"fs-id1165043327626\">As [latex]x\\to 0^+[\/latex], [latex](\\sec^2 x)-2x \\to 1[\/latex] and [latex]x^2 \\sec^2 x+2x \\tan x \\to 0[\/latex]. Since the denominator is positive as [latex]x[\/latex] approaches zero from the right, we conclude that<\/p>\r\n\r\n<div id=\"fs-id1165042710940\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{(\\sec^2 x)-2x}{x^2 \\sec^2 x+2x \\tan x}=\\infty[\/latex].<\/div>\r\n<p id=\"fs-id1165043396304\">Therefore,<\/p>\r\n\r\n<div id=\"fs-id1165043396307\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}(\\frac{1}{x^2}-\\frac{1}{ tan x})=\\infty[\/latex].[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043348549\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165043348552\" class=\"exercise\">\r\n<div id=\"fs-id1165043348554\" class=\"textbox\">\r\n<p id=\"fs-id1165043348557\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim}(\\frac{1}{x}-\\frac{1}{\\sin x})[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043317356\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043317356\"]\r\n<p id=\"fs-id1165043317356\">0<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043317362\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043317368\">Rewrite the difference of fractions as a single fraction.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042364139\">Another type of indeterminate form that arises when evaluating limits involves exponents. The expressions [latex]0^0[\/latex], [latex]\\infty^0[\/latex], and [latex]1^{\\infty}[\/latex] are all indeterminate forms. On their own, these expressions are meaningless because we cannot actually evaluate these expressions as we would evaluate an expression involving real numbers. Rather, these expressions represent forms that arise when finding limits. Now we examine how L\u2019H\u00f4pital\u2019s rule can be used to evaluate limits involving these indeterminate forms.<\/p>\r\n<p id=\"fs-id1165042364178\">Since L\u2019H\u00f4pital\u2019s rule applies to quotients, we use the natural logarithm function and its properties to reduce a problem evaluating a limit involving exponents to a related problem involving a limit of a quotient. For example, suppose we want to evaluate [latex]\\underset{x\\to a}{\\lim}f(x)^{g(x)}[\/latex] and we arrive at the indeterminate form [latex]\\infty^0[\/latex]. (The indeterminate forms [latex]0^0[\/latex] and [latex]1^{\\infty}[\/latex] can be handled similarly.) We proceed as follows. Let<\/p>\r\n\r\n<div id=\"fs-id1165042709633\" class=\"equation unnumbered\">[latex]y=f(x)^{g(x)}[\/latex].<\/div>\r\n<p id=\"fs-id1165043250963\">Then,<\/p>\r\n\r\n<div id=\"fs-id1165043250966\" class=\"equation unnumbered\">[latex]\\ln y=\\ln (f(x)^{g(x)})=g(x) \\ln (f(x))[\/latex].<\/div>\r\n<p id=\"fs-id1165042640888\">Therefore,<\/p>\r\n\r\n<div id=\"fs-id1165042640891\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}[\\ln y]=\\underset{x\\to a}{\\lim}[g(x) \\ln (f(x))][\/latex].<\/div>\r\n<p id=\"fs-id1165043393684\">Since [latex]\\underset{x\\to a}{\\lim}f(x)=\\infty[\/latex], we know that [latex]\\underset{x\\to a}{\\lim}\\ln (f(x))=\\infty[\/latex]. Therefore, [latex]\\underset{x\\to a}{\\lim}g(x) \\ln (f(x))[\/latex] is of the indeterminate form [latex]0 \\cdot \\infty[\/latex], and we can use the techniques discussed earlier to rewrite the expression [latex]g(x) \\ln (f(x))[\/latex] in a form so that we can apply L\u2019H\u00f4pital\u2019s rule. Suppose [latex]\\underset{x\\to a}{\\lim}g(x) \\ln (f(x))=L[\/latex], where [latex]L[\/latex] may be [latex]\\infty [\/latex] or [latex]\u2212\\infty[\/latex]. Then<\/p>\r\n\r\n<div id=\"fs-id1165042638509\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\ln y=L[\/latex].<\/div>\r\n<p id=\"fs-id1165042638551\">Since the natural logarithm function is continuous, we conclude that<\/p>\r\n\r\n<div id=\"fs-id1165042638554\" class=\"equation unnumbered\">[latex]\\ln (\\underset{x\\to a}{\\lim} y)=L[\/latex]<\/div>\r\n<p id=\"fs-id1165042708323\">which gives us<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim} y=\\underset{x\\to a}{\\lim}f(x)^{g(x)}=e^L[\/latex].<\/div>\r\n<div id=\"fs-id1165043390815\" class=\"textbox examples\">\r\n<h3>Indeterminate Form of Type [latex]\\infty^0[\/latex]<\/h3>\r\n<div id=\"fs-id1165043390817\" class=\"exercise\">\r\n<div id=\"fs-id1165043390819\" class=\"textbox\">\r\n<p id=\"fs-id1165043390832\">Evaluate [latex]\\underset{x\\to \\infty }{\\lim} x^{1\/x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043390866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043390866\"]\r\n<p id=\"fs-id1165043390866\">Let [latex]y=x^{1\/x}[\/latex]. Then,<\/p>\r\n\r\n<div id=\"fs-id1165043281565\" class=\"equation unnumbered\">[latex]\\ln (x^{1\/x})=\\frac{1}{x} \\ln x=\\frac{\\ln x}{x}[\/latex].<\/div>\r\n<p id=\"fs-id1165043281615\">We need to evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x}[\/latex]. Applying L\u2019H\u00f4pital\u2019s rule, we obtain<\/p>\r\n\r\n<div id=\"fs-id1165043281645\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim} \\ln y=\\underset{x\\to \\infty}{\\lim}\\frac{\\ln x}{x}=\\underset{x\\to \\infty}{\\lim}\\frac{1\/x}{1}=0[\/latex].<\/div>\r\n<p id=\"fs-id1165043173746\">Therefore, [latex]\\underset{x\\to \\infty }{\\lim}\\ln y=0[\/latex]. Since the natural logarithm function is continuous, we conclude that<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\ln (\\underset{x\\to \\infty}{\\lim} y)=0[\/latex],<\/div>\r\n<p id=\"fs-id1165043427387\">which leads to<\/p>\r\n\r\n<div id=\"fs-id1165043427390\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim} y=\\underset{x\\to \\infty}{\\lim}\\frac{\\ln x}{x}=e^0=1[\/latex].<\/div>\r\n<p id=\"fs-id1165042407320\">Hence,<\/p>\r\n\r\n<div id=\"fs-id1165042407323\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty}{\\lim} x^{1\/x}=1[\/latex].[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042407362\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042407365\" class=\"exercise\">\r\n<div id=\"fs-id1165042407368\" class=\"textbox\">\r\n<p id=\"fs-id1165042407370\">Evaluate [latex]\\underset{x\\to \\infty}{\\lim} x^{1\/ \\ln x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043108248\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043108248\"]\r\n<p id=\"fs-id1165043108248\">[latex]e[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043108254\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165043108260\">Let [latex]y=x^{1\/ \\ln x}[\/latex] and apply the natural logarithm to both sides of the equation.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043108292\" class=\"textbox examples\">\r\n<h3>Indeterminate Form of Type [latex]0^0[\/latex]<\/h3>\r\n<div id=\"fs-id1165043108295\" class=\"exercise\">\r\n<div id=\"fs-id1165043108297\" class=\"textbox\">\r\n<p id=\"fs-id1165042657720\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim} x^{\\sin x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042657755\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042657755\"]\r\n<p id=\"fs-id1165042657755\">Let<\/p>\r\n\r\n<div id=\"fs-id1165042657759\" class=\"equation unnumbered\">[latex]y=x^{\\sin x}[\/latex].<\/div>\r\n<p id=\"fs-id1165042657780\">Therefore,<\/p>\r\n\r\n<div id=\"fs-id1165042657783\" class=\"equation unnumbered\">[latex]\\ln y=\\ln (x^{\\sin x})= \\sin x \\ln x[\/latex].<\/div>\r\n<p id=\"fs-id1165042707171\">We now evaluate [latex]\\underset{x\\to 0^+}{\\lim} \\sin x \\ln x[\/latex]. Since [latex]\\underset{x\\to 0^+}{\\lim} \\sin x=0[\/latex] and [latex]\\underset{x\\to 0^+}{\\lim} \\ln x=\u2212\\infty[\/latex], we have the indeterminate form [latex]0 \\cdot \\infty[\/latex]. To apply L\u2019H\u00f4pital\u2019s rule, we need to rewrite [latex] \\sin x \\ln x[\/latex] as a fraction. We could write<\/p>\r\n\r\n<div id=\"fs-id1165043173832\" class=\"equation unnumbered\">[latex] \\sin x \\ln x=\\frac{\\sin x}{1\/ \\ln x}[\/latex]<\/div>\r\n<p id=\"fs-id1165043173865\">or<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex] \\sin x \\ln x=\\frac{\\ln x}{1\/ \\sin x}=\\frac{\\ln x}{\\csc x}[\/latex].<\/div>\r\n<p id=\"fs-id1165042364489\">Let\u2019s consider the first option. In this case, applying L\u2019H\u00f4pital\u2019s rule, we would obtain<\/p>\r\n\r\n<div id=\"fs-id1165042364494\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim} \\sin x \\ln x=\\underset{x\\to 0^+}{\\lim}\\frac{\\sin x}{1\/ \\ln x}=\\underset{x\\to 0^+}{\\lim}\\frac{\\cos x}{-1\/(x(\\ln x)^2)}=\\underset{x\\to 0^+}{\\lim}(\u2212x(\\ln x)^2 \\cos x)[\/latex].<\/div>\r\n<p id=\"fs-id1165043317274\">Unfortunately, we not only have another expression involving the indeterminate form [latex]0 \\cdot \\infty[\/latex], but the new limit is even more complicated to evaluate than the one with which we started. Instead, we try the second option. By writing<\/p>\r\n\r\n<div id=\"fs-id1165043131555\" class=\"equation unnumbered\">[latex] \\sin x \\ln x=\\frac{\\ln x}{1\/ \\sin x}=\\frac{\\ln x}{\\csc x}[\/latex]<\/div>\r\n<p id=\"fs-id1165043131604\">and applying L\u2019H\u00f4pital\u2019s rule, we obtain<\/p>\r\n\r\n<div id=\"fs-id1165043131609\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim} \\sin x \\ln x=\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{\\csc x}=\\underset{x\\to 0^+}{\\lim}\\frac{1\/x}{\u2212 \\csc x \\cot x}=\\underset{x\\to 0^+}{\\lim}\\frac{-1}{x \\csc x \\cot x}[\/latex].<\/div>\r\n<p id=\"fs-id1165042651533\">Using the fact that [latex]\\csc x=\\frac{1}{\\sin x}[\/latex] and [latex]\\cot x=\\frac{\\cos x}{\\sin x}[\/latex], we can rewrite the expression on the right-hand side as<\/p>\r\n\r\n<div id=\"fs-id1165043251999\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\u2212\\sin^2 x}{x \\cos x}=\\underset{x\\to 0^+}{\\lim}[\\frac{\\sin x}{x} \\cdot (\u2212\\tan x)]=(\\underset{x\\to 0^+}{\\lim}\\frac{\\sin x}{x}) \\cdot (\\underset{x\\to 0^+}{\\lim}(\u2212\\tan x))=1 \\cdot 0=0[\/latex].<\/div>\r\n<p id=\"fs-id1165042676314\">We conclude that [latex]\\underset{x\\to 0^+}{\\lim} \\ln y=0[\/latex]. Therefore, [latex]\\ln (\\underset{x\\to 0^+}{\\lim} y)=0[\/latex] and we have<\/p>\r\n\r\n<div id=\"fs-id1165042327527\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim} y=\\underset{x\\to 0^+}{\\lim} x^{\\sin x}=e^0=1[\/latex].<\/div>\r\n<p id=\"fs-id1165042327592\">Hence,<\/p>\r\n\r\n<div id=\"fs-id1165042327595\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim} x^{\\sin x}=1[\/latex].[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042660254\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042660257\" class=\"exercise\">\r\n<div id=\"fs-id1165042660259\" class=\"textbox\">\r\n<p id=\"fs-id1165042660261\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim} x^x[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042660293\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042660293\"]\r\n<p id=\"fs-id1165042660293\">1<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042660300\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042660306\">Let [latex]y=x^x[\/latex] and take the natural logarithm of both sides of the equation.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042660328\" class=\"bc-section section\">\r\n<h1>Growth Rates of Functions<\/h1>\r\n<p id=\"fs-id1165042660333\">Suppose the functions [latex]f[\/latex] and [latex]g[\/latex] both approach infinity as [latex]x\\to \\infty[\/latex]. Although the values of both functions become arbitrarily large as the values of [latex]x[\/latex] become sufficiently large, sometimes one function is growing more quickly than the other. For example, [latex]f(x)=x^2[\/latex] and [latex]g(x)=x^3[\/latex] both approach infinity as [latex]x\\to \\infty[\/latex]. However, as shown in the following table, the values of [latex]x^3[\/latex] are growing much faster than the values of [latex]x^2[\/latex].<\/p>\r\n\r\n<table id=\"fs-id1165042700402\" class=\"column-header\" summary=\"This table has three rows and five columns. The first column is a header column, and it reads from top to bottom x, f(x) = x2, and g(x) = x3. To the right of the header, the first row reads 10, 100, 1000, and 10,000. The second row reads 100, 10,000, 1,000,000, and 100,000,000. The third row reads 1000, 1,000,000, 1,000,000,000, and 1,000,000,000,000.\"><caption>Comparing the Growth Rates of [latex]x^2[\/latex] and [latex]x^3[\/latex]<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<td>1000<\/td>\r\n<td>10,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]f(x)=x^2[\/latex]<\/strong><\/td>\r\n<td>100<\/td>\r\n<td>10,000<\/td>\r\n<td>1,000,000<\/td>\r\n<td>100,000,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]g(x)=x^3[\/latex]<\/strong><\/td>\r\n<td>1000<\/td>\r\n<td>1,000,000<\/td>\r\n<td>1,000,000,000<\/td>\r\n<td>[latex]1,000,000,000,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165043219185\">In fact,<\/p>\r\n\r\n<div id=\"fs-id1165043219189\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{x^3}{x^2}=\\underset{x\\to \\infty}{\\lim} x=\\infty[\/latex] or, equivalently, [latex]\\underset{x\\to \\infty}{\\lim}\\frac{x^2}{x^3}=\\underset{x\\to \\infty }{\\lim}\\frac{1}{x}=0[\/latex].<\/div>\r\n<p id=\"fs-id1165042465546\">As a result, we say [latex]x^3[\/latex] is growing more rapidly than [latex]x^2[\/latex] as [latex]x\\to \\infty[\/latex]. On the other hand, for [latex]f(x)=x^2[\/latex] and [latex]g(x)=3x^2+4x+1[\/latex], although the values of [latex]g(x)[\/latex] are always greater than the values of [latex]f(x)[\/latex] for [latex]x&gt;0[\/latex], each value of [latex]g(x)[\/latex] is roughly three times the corresponding value of [latex]f(x)[\/latex] as [latex]x\\to \\infty[\/latex], as shown in the following table. In fact,<\/p>\r\n\r\n<div id=\"fs-id1165042461121\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{x^2}{3x^2+4x+1}=\\frac{1}{3}[\/latex].<\/div>\r\n<table id=\"fs-id1165042461176\" class=\"column-header\" summary=\"This table has three rows and five columns. The first column is a header column, and it reads from top to bottom x, f(x) = x2, and g(x) = 3x2 + 4x + 1. To the right of the header, the first row reads 10, 100, 1000, and 10,000. The second row reads 100, 10,000, 1,000,000, and 100,000,000. The third row reads 341, 30,401, 3,004,001, and 300,040,001.\"><caption>Comparing the Growth Rates of [latex]x^2[\/latex] and [latex]3x^2+4x+1[\/latex]<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<td>1000<\/td>\r\n<td>10,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]f(x)=x^2[\/latex]<\/strong><\/td>\r\n<td>100<\/td>\r\n<td>10,000<\/td>\r\n<td>1,000,000<\/td>\r\n<td>100,000,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]g(x)=3x^2+4x+1[\/latex]<\/strong><\/td>\r\n<td>341<\/td>\r\n<td>30,401<\/td>\r\n<td>3,004,001<\/td>\r\n<td>300,040,001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042659418\">In this case, we say that [latex]x^2[\/latex] and [latex]3x^2+4x+1[\/latex] are growing at the same rate as [latex]x\\to \\infty[\/latex].<\/p>\r\n<p id=\"fs-id1165042659463\">More generally, suppose [latex]f[\/latex] and [latex]g[\/latex] are two functions that approach infinity as [latex]x\\to \\infty[\/latex]. We say [latex]g[\/latex] grows more rapidly than [latex]f[\/latex] as [latex]x\\to \\infty [\/latex] if<\/p>\r\n\r\n<div id=\"fs-id1165042659507\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{g(x)}{f(x)}=\\infty[\/latex] or, equivalently, [latex]\\underset{x\\to \\infty }{\\lim}\\frac{f(x)}{g(x)}=0[\/latex].<\/div>\r\n<p id=\"fs-id1165043422372\">On the other hand, if there exists a constant [latex]M \\ne 0[\/latex] such that<\/p>\r\n\r\n<div id=\"fs-id1165043422384\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{f(x)}{g(x)}=M[\/latex],<\/div>\r\n<p id=\"fs-id1165043422432\">we say [latex]f[\/latex] and [latex]g[\/latex] grow at the same rate as [latex]x\\to \\infty[\/latex].<\/p>\r\n<p id=\"fs-id1165043422456\">Next we see how to use L\u2019H\u00f4pital\u2019s rule to compare the growth rates of power, exponential, and logarithmic functions.<\/p>\r\n\r\n<div id=\"fs-id1165043422462\" class=\"textbox examples\">\r\n<h3>Comparing the Growth Rates of [latex]\\ln x[\/latex], [latex]x^2[\/latex], and [latex]e^x[\/latex]<\/h3>\r\n<div id=\"fs-id1165043422464\" class=\"exercise\">\r\n<div id=\"fs-id1165043422466\" class=\"textbox\">\r\n<p id=\"fs-id1165042325994\">For each of the following pairs of functions, use L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to \\infty }{\\lim}(\\frac{f(x)}{g(x)})[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1165042326044\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]f(x)=x^2[\/latex] and [latex]g(x)=e^x[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\ln x[\/latex] and [latex]g(x)=x^2[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042326147\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042326147\"]\r\n<ol id=\"fs-id1165042326147\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Since [latex]\\underset{x\\to \\infty }{\\lim} x^2=\\infty [\/latex] and [latex]\\underset{x\\to \\infty }{\\lim} e^x[\/latex], we can use L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to \\infty }{\\lim}[\\frac{x^2}{e^x}][\/latex]. We obtain\r\n<div id=\"fs-id1165042631803\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{x^2}{e^x}=\\underset{x\\to \\infty }{\\lim}\\frac{2x}{e^x}[\/latex].<\/div>\r\nSince [latex]\\underset{x\\to \\infty }{\\lim}2x=\\infty [\/latex] and [latex]\\underset{x\\to \\infty }{\\lim}e^x=\\infty[\/latex], we can apply L\u2019H\u00f4pital\u2019s rule again. Since\r\n<div id=\"fs-id1165042631923\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{2x}{e^x}=\\underset{x\\to \\infty }{\\lim}\\frac{2}{e^x}=0[\/latex],<\/div>\r\nwe conclude that\r\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{x^2}{e^x}=0[\/latex].<\/div>\r\nTherefore, [latex]e^x[\/latex] grows more rapidly than [latex]x^2[\/latex] as [latex]x\\to \\infty [\/latex] (See <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_08_003\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165042418200\">(Figure)<\/a>).\r\n<div id=\"CNX_Calc_Figure_04_08_003\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"271\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211309\/CNX_Calc_Figure_04_08_005.jpg\" alt=\"The functions g(x) = ex and f(x) = x2 are graphed. It is obvious that g(x) increases much more quickly than f(x).\" width=\"271\" height=\"278\" \/> <strong>Figure 3.<\/strong> An exponential function grows at a faster rate than a power function.[\/caption]\r\n\r\n<\/div>\r\n<table id=\"fs-id1165042418200\" class=\"column-header\" summary=\"This table has three rows and five columns. The first column is a header column, and it reads from top to bottom x, x2, and ex. To the right of the header, the first row reads 5, 10, 15, and 20. The second row reads 25, 100, 225, and 400. The third row reads 148, 22,026, 3,269,017, and 485,165,195.\"><caption>Growth rates of a power function and an exponential function.<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>5<\/td>\r\n<td>10<\/td>\r\n<td>15<\/td>\r\n<td>20<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x^2[\/latex]<\/strong><\/td>\r\n<td>25<\/td>\r\n<td>100<\/td>\r\n<td>225<\/td>\r\n<td>400<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]e^x[\/latex]<\/strong><\/td>\r\n<td>148<\/td>\r\n<td>22,026<\/td>\r\n<td>3,269,017<\/td>\r\n<td>485,165,195<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Since [latex]\\underset{x\\to \\infty }{\\lim} \\ln x=\\infty [\/latex] and [latex]\\underset{x\\to \\infty }{\\lim} x^2=\\infty[\/latex], we can use L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x^2}[\/latex]. We obtain\r\n<div id=\"fs-id1165042471196\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x^2}=\\underset{x\\to \\infty }{\\lim}\\frac{1\/x}{2x}=\\underset{x\\to \\infty }{\\lim}\\frac{1}{2x^2}=0[\/latex].<\/div>\r\nThus, [latex]x^2[\/latex] grows more rapidly than [latex]\\ln x[\/latex] as [latex]x\\to \\infty [\/latex] (see <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_08_004\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165042471354\">(Figure)<\/a>).\r\n<div id=\"CNX_Calc_Figure_04_08_004\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211312\/CNX_Calc_Figure_04_08_006.jpg\" alt=\"The functions g(x) = x2 and f(x) = ln(x) are graphed. It is obvious that g(x) increases much more quickly than f(x).\" width=\"417\" height=\"347\" \/> <strong>Figure 4.<\/strong> A power function grows at a faster rate than a logarithmic function.[\/caption]\r\n\r\n<\/div>\r\n<table id=\"fs-id1165042471354\" class=\"column-header\" summary=\"This table has three rows and five columns. The first column is a header column, and it reads from top to bottom x, ln(x), and x2. To the right of the header, the first row reads 10, 100, 1000, and 10,000. The second row reads 2.303, 4.605, 6.908, and 9.210. The third row reads 100, 10,000, 1,000,000, and 100,000,000.\"><caption>Growth rates of a power function and a logarithmic function<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<td>1000<\/td>\r\n<td>10,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]\\ln x[\/latex]<\/strong><\/td>\r\n<td>2.303<\/td>\r\n<td>4.605<\/td>\r\n<td>6.908<\/td>\r\n<td>9.210<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x^2[\/latex]<\/strong><\/td>\r\n<td>100<\/td>\r\n<td>10,000<\/td>\r\n<td>1,000,000<\/td>\r\n<td>100,000,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042463715\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1165042463719\" class=\"exercise\">\r\n<div id=\"fs-id1165042463721\" class=\"textbox\">\r\n<p id=\"fs-id1165042463723\">Compare the growth rates of [latex]x^{100}[\/latex] and [latex]2^x[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042463749\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042463749\"]\r\n<p id=\"fs-id1165042463749\">The function [latex]2^x[\/latex] grows faster than [latex]x^{100}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042463772\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1165042463780\">Apply L\u2019H\u00f4pital\u2019s rule to [latex]x^{100}\/2^x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042463803\">Using the same ideas as in <a class=\"autogenerated-content\" href=\"#fs-id1165043422462\">(Figure)<\/a>a. it is not difficult to show that [latex]e^x[\/latex] grows more rapidly than [latex]x^p[\/latex] for any [latex]p&gt;0[\/latex]. In <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_08_005\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165043327297\">(Figure)<\/a>, we compare [latex]e^x[\/latex] with [latex]x^3[\/latex] and [latex]x^4[\/latex] as [latex]x\\to \\infty[\/latex].<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_08_005\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"858\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211316\/CNX_Calc_Figure_04_08_001.jpg\" alt=\"This figure has two figures marked a and b. In figure a, the functions y = ex and y = x3 are graphed. It is obvious that ex increases more quickly than x3. In figure b, the functions y = ex and y = x4 are graphed. It is obvious that ex increases much more quickly than x4, but the point at which that happens is further to the right than it was for x3.\" width=\"858\" height=\"386\" \/> <strong>Figure 5.<\/strong> The exponential function [latex]e^x[\/latex] grows faster than [latex]x^p[\/latex] for any [latex]p&gt;0[\/latex]. (a) A comparison of [latex]e^x[\/latex] with [latex]x^3[\/latex]. (b) A comparison of [latex]e^x[\/latex] with [latex]x^4[\/latex].[\/caption]<\/div>\r\n<table class=\"column-header\" summary=\"This table has four rows and five columns. The first column is a header column, and it reads from top to bottom x, x3, x4, and ex. To the right of the header, the first row reads 5, 10, 15, and 20. The second row reads 125, 1000, 3375, and 8000. The third row reads 625, 10,000, 50,625, and 160,000. The fourth row reads 148, 22,026, 3,269,017, and 485,165,195.\"><caption>An exponential function grows at a faster rate than any power function<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>5<\/td>\r\n<td>10<\/td>\r\n<td>15<\/td>\r\n<td>20<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x^3[\/latex]<\/strong><\/td>\r\n<td>125<\/td>\r\n<td>1000<\/td>\r\n<td>3375<\/td>\r\n<td>8000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x^4[\/latex]<\/strong><\/td>\r\n<td>625<\/td>\r\n<td>10,000<\/td>\r\n<td>50,625<\/td>\r\n<td>160,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]e^x[\/latex]<\/strong><\/td>\r\n<td>148<\/td>\r\n<td>22,026<\/td>\r\n<td>3,269,017<\/td>\r\n<td>485,165,195<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042542902\">Similarly, it is not difficult to show that [latex]x^p[\/latex] grows more rapidly than [latex]\\ln x[\/latex] for any [latex]p&gt;0[\/latex]. In <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_08_006\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165042460279\">(Figure)<\/a>, we compare [latex]\\ln x[\/latex] with [latex]\\sqrt[3]{x}[\/latex] and [latex]\\sqrt{x}[\/latex].<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_04_08_006\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"343\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211319\/CNX_Calc_Figure_04_08_007.jpg\" alt=\"This figure shows y = the square root of x, y = the cube root of x, and y = ln(x). It is apparent that y = ln(x) grows more slowly than either of these functions.\" width=\"343\" height=\"203\" \/> <strong>Figure 6.<\/strong> The function [latex]y=\\ln x[\/latex] grows more slowly than [latex]x^p[\/latex] for any [latex]p&gt;0[\/latex] as [latex]x\\to \\infty[\/latex].[\/caption]<\/div>\r\n<table id=\"fs-id1165042460279\" class=\"column-header\" summary=\"This table has four rows and five columns. The first column is a header column, and it reads from top to bottom x, ln(x), the cube root of x, and the square root of x. To the right of the header, the first row reads 10, 100, 1000, and 10,000. The second row reads 2.303, 4.605, 6.908, and 9.210. The third row reads 2.154, 4.642, 10, and 21.544. The fourth row reads 3.162, 10, 31.623, and 100.\"><caption>A logarithmic function grows at a slower rate than any root function<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>100<\/td>\r\n<td>1000<\/td>\r\n<td>10,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]\\ln x[\/latex]<\/strong><\/td>\r\n<td>2.303<\/td>\r\n<td>4.605<\/td>\r\n<td>6.908<\/td>\r\n<td>9.210<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]\\sqrt[3]{x}[\/latex]<\/strong><\/td>\r\n<td>2.154<\/td>\r\n<td>4.642<\/td>\r\n<td>10<\/td>\r\n<td>21.544<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]\\sqrt{x}[\/latex]<\/strong><\/td>\r\n<td>3.162<\/td>\r\n<td>10<\/td>\r\n<td>31.623<\/td>\r\n<td>100<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165042658525\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165042658532\">\r\n \t<li>L\u2019H\u00f4pital\u2019s rule can be used to evaluate the limit of a quotient when the indeterminate form [latex]0\/0[\/latex] or [latex]\\infty \/ \\infty[\/latex] arises.<\/li>\r\n \t<li>L\u2019H\u00f4pital\u2019s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form [latex]0\/0[\/latex] or [latex]\\infty \/ \\infty[\/latex].<\/li>\r\n \t<li>The exponential function [latex]e^x[\/latex] grows faster than any power function [latex]x^p[\/latex], [latex]p&gt;0[\/latex].<\/li>\r\n \t<li>The logarithmic function [latex]\\ln x[\/latex] grows more slowly than any power function [latex]x^p[\/latex], [latex]p&gt;0[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165042658654\" class=\"textbox exercises\">\r\n<p id=\"fs-id1165043217918\">For the following exercises, evaluate the limit.<\/p>\r\n\r\n<div id=\"fs-id1165043217921\" class=\"exercise\">\r\n<div id=\"fs-id1165043217924\" class=\"textbox\">\r\n<p id=\"fs-id1165043217926\"><strong>1.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043217966\" class=\"exercise\">\r\n<div id=\"fs-id1165043217969\" class=\"textbox\">\r\n<p id=\"fs-id1165043217971\"><strong>2.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x^k}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043218009\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043218009\"]\r\n<p id=\"fs-id1165043218009\">[latex]\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043218017\" class=\"exercise\">\r\n<div id=\"fs-id1165043218019\" class=\"textbox\">\r\n<p id=\"fs-id1165043218021\"><strong>3.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x^k}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043218066\" class=\"exercise\">\r\n<div id=\"fs-id1165043218068\" class=\"textbox\">\r\n<p id=\"fs-id1165043218070\"><strong>4.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^2-a^2}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043218117\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043218117\"]\r\n<p id=\"fs-id1165043218117\">[latex]\\frac{1}{2a}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043218132\" class=\"exercise\">\r\n<div id=\"fs-id1165043218134\" class=\"textbox\">\r\n<p id=\"fs-id1165043218136\"><strong>5.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^3-a^3}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042469654\" class=\"exercise\">\r\n<div id=\"fs-id1165042469656\" class=\"textbox\">\r\n<p id=\"fs-id1165042469659\"><strong>6.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^n-a^n}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042469706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042469706\"]\r\n<p id=\"fs-id1165042469706\">[latex]\\frac{1}{na^{n-1}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042469729\">For the following exercises, determine whether you can apply L\u2019H\u00f4pital\u2019s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L\u2019H\u00f4pital\u2019s rule.<\/p>\r\n\r\n<div id=\"fs-id1165042469737\" class=\"exercise\">\r\n<div id=\"fs-id1165042469739\" class=\"textbox\">\r\n<p id=\"fs-id1165042469741\"><strong>7.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim}x^2 \\ln x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042469817\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n<strong>8.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x^{1\/x}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042711556\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711556\"]\r\n<p id=\"fs-id1165042711556\">Cannot apply directly; use logarithms<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042711562\" class=\"exercise\">\r\n<div id=\"fs-id1165042711564\" class=\"textbox\">\r\n<p id=\"fs-id1165042711566\"><strong>9.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{2\/x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042711603\" class=\"exercise\">\r\n<div id=\"fs-id1165042711605\" class=\"textbox\">\r\n<p id=\"fs-id1165042711607\"><strong>10.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{x^2}{1\/x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042711643\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711643\"]\r\n<p id=\"fs-id1165042711643\">Cannot apply directly; rewrite as [latex]\\underset{x\\to 0}{\\lim} x^3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042711670\" class=\"exercise\">\r\n<div id=\"fs-id1165042711672\" class=\"textbox\">\r\n<p id=\"fs-id1165042711674\"><strong>11.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042711711\">For the following exercises, evaluate the limits with either L\u2019H\u00f4pital\u2019s rule or previously learned methods.<\/p>\r\n\r\n<div id=\"fs-id1165042711715\" class=\"exercise\">\r\n<div id=\"fs-id1165042711717\" class=\"textbox\">\r\n<p id=\"fs-id1165042711719\"><strong>12.<\/strong> [latex]\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x-3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042711760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711760\"]\r\n<p id=\"fs-id1165042711760\">6<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042711767\" class=\"exercise\">\r\n<div id=\"fs-id1165042711769\" class=\"textbox\">\r\n<p id=\"fs-id1165042711771\"><strong>13.<\/strong> [latex]\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x+3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042499499\" class=\"exercise\">\r\n<div id=\"fs-id1165042499502\" class=\"textbox\">\r\n<p id=\"fs-id1165042499504\"><strong>14.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^{-2}-1}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042499552\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042499552\"]\r\n<p id=\"fs-id1165042499552\">-2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042499561\" class=\"exercise\">\r\n<div id=\"fs-id1165042499563\" class=\"textbox\">\r\n<p id=\"fs-id1165042499565\"><strong>15.<\/strong> [latex]\\underset{x\\to \\pi \/2}{\\lim}\\frac{\\cos x}{\\pi \/ 2 - x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042499616\" class=\"exercise\">\r\n<div id=\"fs-id1165042499618\" class=\"textbox\">\r\n<p id=\"fs-id1165042499620\"><strong>16.<\/strong> [latex]\\underset{x\\to \\pi }{\\lim}\\frac{x-\\pi }{\\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042499655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042499655\"]\r\n<p id=\"fs-id1165042499655\">-1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042499664\" class=\"exercise\">\r\n<div id=\"fs-id1165042499666\" class=\"textbox\">\r\n<p id=\"fs-id1165042499668\"><strong>17.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{\\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042711252\" class=\"exercise\">\r\n<div id=\"fs-id1165042711254\" class=\"textbox\">\r\n<p id=\"fs-id1165042711256\"><strong>18.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^n-1}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042711303\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711303\"]\r\n<p id=\"fs-id1165042711303\">[latex]n[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n<strong>19.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^n-1-nx}{x^2}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042711396\" class=\"exercise\">\r\n<div id=\"fs-id1165042711398\" class=\"textbox\">\r\n<p id=\"fs-id1165042711401\"><strong>20.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x- \\tan x}{x^3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042711441\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042711441\"]\r\n<p id=\"fs-id1165042711441\">[latex]-\\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1165042711459\"><strong>21.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sqrt{1+x}-\\sqrt{1-x}}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042602902\" class=\"exercise\">\r\n<div id=\"fs-id1165042602904\" class=\"textbox\">\r\n<p id=\"fs-id1165042602906\"><strong>22.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-x-1}{x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042602950\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042602950\"]\r\n<p id=\"fs-id1165042602950\">[latex]\\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042602961\" class=\"exercise\">\r\n<div id=\"fs-id1165042602964\" class=\"textbox\">\r\n<p id=\"fs-id1165042602966\"><strong>23.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\tan x}{\\sqrt{x}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042603005\" class=\"exercise\">\r\n<div id=\"fs-id1165042603007\" class=\"textbox\">\r\n<p id=\"fs-id1165042603009\"><strong>24.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{\\ln x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042603044\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042603044\"]\r\n<p id=\"fs-id1165042603044\">1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042603052\" class=\"exercise\">\r\n<div id=\"fs-id1165042603054\" class=\"textbox\">\r\n<p id=\"fs-id1165042603056\"><strong>25.<\/strong> [latex]\\underset{x\\to 0}{\\lim}(x+1)^{1\/x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042603107\" class=\"exercise\">\r\n<div id=\"fs-id1165042603109\" class=\"textbox\">\r\n<p id=\"fs-id1165042603111\"><strong>26.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{\\sqrt{x}-\\sqrt[3]{x}}{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042617540\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042617540\"]\r\n<p id=\"fs-id1165042617540\">[latex]\\frac{1}{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042617552\" class=\"exercise\">\r\n<div id=\"fs-id1165042617554\" class=\"textbox\">\r\n<p id=\"fs-id1165042617556\"><strong>27.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim} x^{2x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042617597\" class=\"exercise\">\r\n<div id=\"fs-id1165042617599\" class=\"textbox\">\r\n<p id=\"fs-id1165042617601\"><strong>28.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x \\sin (\\frac{1}{x})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042617638\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042617638\"]\r\n<p id=\"fs-id1165042617638\">1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042617646\" class=\"exercise\">\r\n<div id=\"fs-id1165042617648\" class=\"textbox\">\r\n<p id=\"fs-id1165042617650\"><strong>29.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042617696\" class=\"exercise\">\r\n<div id=\"fs-id1165042617698\" class=\"textbox\">\r\n<p id=\"fs-id1165042617700\"><strong>30.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim} x \\ln (x^4)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042617740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042617740\"]\r\n<p id=\"fs-id1165042617740\">0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042617748\" class=\"exercise\">\r\n<div id=\"fs-id1165042617750\" class=\"textbox\">\r\n<p id=\"fs-id1165042617752\"><strong>31.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}(x-e^x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042707732\" class=\"exercise\">\r\n<div id=\"fs-id1165042707734\" class=\"textbox\">\r\n<p id=\"fs-id1165042707736\"><strong>32.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x^2 e^{\u2212x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">[reveal-answer q=\"849216\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"849216\"]0[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042707778\" class=\"exercise\">\r\n<div id=\"fs-id1165042707780\" class=\"textbox\">\r\n<p id=\"fs-id1165042707782\"><strong>33.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{3^x-2^x}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042707843\" class=\"exercise\">\r\n<div id=\"fs-id1165042707845\" class=\"textbox\">\r\n\r\n<strong>34.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{1+1\/x}{1-1\/x}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042707893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042707893\"]\r\n<p id=\"fs-id1165042707893\">-1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1165042707904\" class=\"textbox\">\r\n<p id=\"fs-id1165042707906\"><strong>35.<\/strong> [latex]\\underset{x\\to \\pi \/4}{\\lim}(1- \\tan x) \\cot x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042707959\" class=\"exercise\">\r\n<div id=\"fs-id1165042707961\" class=\"textbox\">\r\n<p id=\"fs-id1165042707963\"><strong>36.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} xe^{1\/x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042525331\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042525331\"]\r\n<p id=\"fs-id1165042525331\">[latex]\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042525338\" class=\"exercise\">\r\n<div id=\"fs-id1165042525340\" class=\"textbox\">\r\n<p id=\"fs-id1165042525342\"><strong>37.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{1\/ \\cos x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042525384\" class=\"exercise\">\r\n<div id=\"fs-id1165042525386\" class=\"textbox\">\r\n<p id=\"fs-id1165042525388\"><strong>38.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{1\/x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042525418\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042525418\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042525418\"]1[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042525428\" class=\"exercise\">\r\n<div id=\"fs-id1165042525430\" class=\"textbox\">\r\n<p id=\"fs-id1165042525432\"><strong>39.<\/strong> [latex]\\underset{x\\to 0}{\\lim} (1-\\frac{1}{x})^x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042525481\" class=\"exercise\">\r\n<div id=\"fs-id1165042525483\" class=\"textbox\">\r\n<p id=\"fs-id1165042525485\"><strong>40.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} (1-\\frac{1}{x})^x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042525526\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042525526\"]\r\n<p id=\"fs-id1165042525526\">[latex]\\frac{1}{e}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042525538\">For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L\u2019H\u00f4pital\u2019s rule to find the limit directly.<\/p>\r\n\r\n<div id=\"fs-id1165042525544\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\n<strong>41. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-1}{x}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043382933\" class=\"exercise\">\r\n<div id=\"fs-id1165043382935\" class=\"textbox\">\r\n<p id=\"fs-id1165043382937\"><strong>42. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x \\sin (\\frac{1}{x})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165043382979\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043382979\"]\r\n<p id=\"fs-id1165043382979\">0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043382987\" class=\"exercise\">\r\n<div id=\"fs-id1165043382989\" class=\"textbox\">\r\n<p id=\"fs-id1165043382991\"><strong>43. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{1- \\cos (\\pi x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043383052\" class=\"exercise\">\r\n<div id=\"fs-id1165043383054\" class=\"textbox\">\r\n<p id=\"fs-id1165043383056\"><strong>44. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{e^{x-1}-1}{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<div class=\"textbox shaded\">[reveal-answer q=\"855100\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"855100\"]1[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043383121\" class=\"exercise\">\r\n<div id=\"fs-id1165043383123\" class=\"textbox\">\r\n<p id=\"fs-id1165043383125\"><strong>45. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{(x-1)^2}{\\ln x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042710174\" class=\"exercise\">\r\n<div id=\"fs-id1165042710176\" class=\"textbox\">\r\n<p id=\"fs-id1165042710178\"><strong>46. [T]\u00a0<\/strong>[latex]\\underset{x\\to \\pi }{\\lim}\\frac{1+ \\cos x}{ \\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042710220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042710220\"]\r\n<p id=\"fs-id1165042710220\">0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042710228\" class=\"exercise\">\r\n<div id=\"fs-id1165042710230\" class=\"textbox\">\r\n<p id=\"fs-id1165042710232\"><strong>47. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}( \\csc x-\\frac{1}{x})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042710284\" class=\"exercise\">\r\n<div id=\"fs-id1165042710286\" class=\"textbox\">\r\n<p id=\"fs-id1165042710288\"><strong>48. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim} \\tan (x^x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042710332\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042710332\"]\r\n<p id=\"fs-id1165042710332\">[latex] \\tan (1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042710348\" class=\"exercise\">\r\n<div id=\"fs-id1165042710350\" class=\"textbox\">\r\n<p id=\"fs-id1165042710352\"><strong>49. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{ \\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042710404\" class=\"exercise\">\r\n<div id=\"fs-id1165042710406\" class=\"textbox\">\r\n<p id=\"fs-id1165042710408\"><strong>50. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-e^{\u2212x}}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165042539145\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042539145\"]\r\n<p id=\"fs-id1165042539145\">2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165042539157\" class=\"definition\">\r\n \t<dt>indeterminate forms<\/dt>\r\n \t<dd id=\"fs-id1165042539162\">when evaluating a limit, the forms [latex]0\/0[\/latex], [latex]\\infty \/ \\infty[\/latex], [latex]0 \\cdot \\infty[\/latex], [latex]\\infty -\\infty[\/latex], [latex]0^0[\/latex], [latex]\\infty^0[\/latex], and [latex]1^{\\infty}[\/latex] are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042539243\" class=\"definition\">\r\n \t<dt>L\u2019H\u00f4pital\u2019s rule<\/dt>\r\n \t<dd id=\"fs-id1165042539249\">if [latex]f[\/latex] and [latex]g[\/latex] are differentiable functions over an interval [latex]a[\/latex], except possibly at [latex]a[\/latex], and [latex]\\underset{x\\to a}{\\lim} f(x)=0=\\underset{x\\to a}{\\lim} g(x)[\/latex] or [latex]\\underset{x\\to a}{\\lim} f(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim} g(x)[\/latex] are infinite, then [latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex], assuming the limit on the right exists or is [latex]\\infty [\/latex] or [latex]\u2212\\infty [\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Recognize when to apply L\u2019H\u00f4pital\u2019s rule.<\/li>\n<li>Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L\u2019H\u00f4pital\u2019s rule in each case.<\/li>\n<li>Describe the relative growth rates of functions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165042709572\">In this section, we examine a powerful tool for evaluating limits. This tool, known as <strong>L\u2019H\u00f4pital\u2019s rule<\/strong>, uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.<\/p>\n<div id=\"fs-id1165043085155\" class=\"bc-section section\">\n<h1>Applying L\u2019H\u00f4pital\u2019s Rule<\/h1>\n<p id=\"fs-id1165042941863\">L\u2019H\u00f4pital\u2019s rule can be used to evaluate limits involving the quotient of two functions. Consider<\/p>\n<div id=\"fs-id1165042458008\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}[\/latex].<\/div>\n<p id=\"fs-id1165042613148\">If [latex]\\underset{x\\to a}{\\lim}f(x)=L_1[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=L_2 \\ne 0[\/latex], then<\/p>\n<div id=\"fs-id1165043088102\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\frac{L_1}{L_2}[\/latex].<\/div>\n<p id=\"fs-id1165042330308\">However, what happens if [latex]\\underset{x\\to a}{\\lim}f(x)=0[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=0[\/latex]? We call this one of the <strong>indeterminate forms<\/strong>, of type [latex]\\frac{0}{0}[\/latex]. This is considered an indeterminate form because we cannot determine the exact behavior of [latex]\\frac{f(x)}{g(x)}[\/latex] as [latex]x\\to a[\/latex] without further analysis. We have seen examples of this earlier in the text. For example, consider<\/p>\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to 2}{\\lim}\\frac{x^2-4}{x-2}[\/latex] and [latex]\\underset{x\\to 0}{\\lim}\\frac{ \\sin x}{x}[\/latex].<\/div>\n<p id=\"fs-id1165043036412\">For the first of these examples, we can evaluate the limit by factoring the numerator and writing<\/p>\n<div id=\"fs-id1165043067705\" class=\"equation unnumbered\">[latex]\\underset{x\\to 2}{\\lim}\\frac{x^2-4}{x-2}=\\underset{x\\to 2}{\\lim}\\frac{(x+2)(x-2)}{x-2}=\\underset{x\\to 2}{\\lim}(x+2)=2+2=4[\/latex].<\/div>\n<p>For [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}[\/latex] we were able to show, using a geometric argument, that<\/p>\n<div id=\"fs-id1165042954700\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}=1[\/latex].<\/div>\n<p id=\"fs-id1165043001194\">Here we use a different technique for evaluating limits such as these. Not only does this technique provide an easier way to evaluate these limits, but also, and more important, it provides us with a way to evaluate many other limits that we could not calculate previously.<\/p>\n<p id=\"fs-id1165043062373\">The idea behind L\u2019H\u00f4pital\u2019s rule can be explained using local linear approximations. Consider two differentiable functions [latex]f[\/latex] and [latex]g[\/latex] such that [latex]\\underset{x\\to a}{\\lim}f(x)=0=\\underset{x\\to a}{\\lim}g(x)[\/latex] and such that [latex]g^{\\prime}(a)\\ne 0[\/latex] For [latex]x[\/latex] near [latex]a[\/latex], we can write<\/p>\n<div id=\"fs-id1165043199940\" class=\"equation unnumbered\">[latex]f(x)\\approx f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/div>\n<p id=\"fs-id1165042355300\">and<\/p>\n<div id=\"fs-id1165042320393\" class=\"equation unnumbered\">[latex]g(x)\\approx g(a)+g^{\\prime}(a)(x-a)[\/latex].<\/div>\n<p id=\"fs-id1165043178271\">Therefore,<\/p>\n<div id=\"fs-id1165042331440\" class=\"equation unnumbered\">[latex]\\frac{f(x)}{g(x)}\\approx \\frac{f(a)+f^{\\prime}(a)(x-a)}{g(a)+g^{\\prime}(a)(x-a)}[\/latex].<\/div>\n<div id=\"CNX_Calc_Figure_04_08_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 628px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211304\/CNX_Calc_Figure_04_08_003.jpg\" alt=\"Two functions y = f(x) and y = g(x) are drawn such that they cross at a point above x = a. The linear approximations of these two functions y = f(a) + f\u2019(a)(x \u2013 a) and y = g(a) + g\u2019(a)(x \u2013 a) are also drawn.\" width=\"618\" height=\"390\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.<\/strong> If [latex]\\underset{x\\to a}{\\lim}f(x)=\\underset{x\\to a}{\\lim}g(x)[\/latex], then the ratio [latex]f(x)\/g(x)[\/latex] is approximately equal to the ratio of their linear approximations near [latex]a[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>Since [latex]f[\/latex] is differentiable at [latex]a[\/latex], then [latex]f[\/latex] is continuous at [latex]a[\/latex], and therefore [latex]f(a)=\\underset{x\\to a}{\\lim}f(x)=0[\/latex]. Similarly, [latex]g(a)=\\underset{x\\to a}{\\lim}g(x)=0[\/latex]. If we also assume that [latex]f^{\\prime}[\/latex] and [latex]g^{\\prime}[\/latex] are continuous at [latex]x=a[\/latex], then [latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}f^{\\prime}(x)[\/latex] and [latex]g^{\\prime}(a)=\\underset{x\\to a}{\\lim}g^{\\prime}(x)[\/latex]. Using these ideas, we conclude that<\/p>\n<div id=\"fs-id1165042373953\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)(x-a)}{g^{\\prime}(x)(x-a)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex].<\/div>\n<p>Note that the assumption that [latex]f^{\\prime}[\/latex] and [latex]g^{\\prime}[\/latex] are continuous at [latex]a[\/latex] and [latex]g^{\\prime}(a)\\ne 0[\/latex] can be loosened. We state L\u2019H\u00f4pital\u2019s rule formally for the indeterminate form [latex]\\frac{0}{0}[\/latex]. Also note that the notation [latex]\\frac{0}{0}[\/latex] does not mean we are actually dividing zero by zero. Rather, we are using the notation [latex]\\frac{0}{0}[\/latex] to represent a quotient of limits, each of which is zero.<\/p>\n<div id=\"fs-id1165043352593\" class=\"textbox key-takeaways theorem\">\n<h3>L\u2019H\u00f4pital\u2019s Rule (0\/0 Case)<\/h3>\n<p id=\"fs-id1165043276140\">Suppose [latex]f[\/latex] and [latex]g[\/latex] are differentiable functions over an open interval containing [latex]a[\/latex], except possibly at [latex]a[\/latex]. If [latex]\\underset{x\\to a}{\\lim}f(x)=0[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=0[\/latex], then<\/p>\n<div id=\"fs-id1165043020148\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex],<\/div>\n<p id=\"fs-id1165043035673\">assuming the limit on the right exists or is [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex]. This result also holds if we are considering one-sided limits, or if [latex]a=\\infty[\/latex] or [latex]-\\infty[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165042712928\" class=\"bc-section section\">\n<h2>Proof<\/h2>\n<p id=\"fs-id1165043429940\">We provide a proof of this theorem in the special case when [latex]f, \\, g, \\, f^{\\prime}[\/latex], and [latex]g^{\\prime}[\/latex] are all continuous over an open interval containing [latex]a[\/latex]. In that case, since [latex]\\underset{x\\to a}{\\lim}f(x)=0=\\underset{x\\to a}{\\lim}g(x)[\/latex] and [latex]f[\/latex] and [latex]g[\/latex] are continuous at [latex]a[\/latex], it follows that [latex]f(a)=0=g(a)[\/latex]. Therefore,<\/p>\n<div id=\"fs-id1165043353664\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll} \\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)} & =\\underset{x\\to a}{\\lim}\\frac{f(x)-f(a)}{g(x)-g(a)} & & & \\text{since} \\, f(a)=0=g(a) \\\\ & =\\underset{x\\to a}{\\lim}\\frac{\\frac{f(x)-f(a)}{x-a}}{\\frac{g(x)-g(a)}{x-a}} & & & \\text{multiplying numerator and denominator by} \\, \\frac{1}{x-a} \\\\ & =\\frac{\\underset{x\\to a}{\\lim}\\frac{f(x)-f(a)}{x-a}}{\\underset{x\\to a}{\\lim}\\frac{g(x)-g(a)}{x-a}} & & & \\text{limit of a quotient} \\\\ & =\\frac{f^{\\prime}(a)}{g^{\\prime}(a)} & & & \\text{definition of the derivative} \\\\ & =\\frac{\\underset{x\\to a}{\\lim}f^{\\prime}(x)}{\\underset{x\\to a}{\\lim}g^{\\prime}(x)} & & & \\text{continuity of} \\, f^{\\prime} \\, \\text{and} \\, g^{\\prime} \\\\ & =\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)} & & & \\text{limit of a quotient} \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165043092431\">Note that L\u2019H\u00f4pital\u2019s rule states we can calculate the limit of a quotient [latex]\\frac{f}{g}[\/latex] by considering the limit of the quotient of the derivatives [latex]\\frac{f^{\\prime}}{g^{\\prime}}[\/latex]. It is important to realize that we are not calculating the derivative of the quotient [latex]\\frac{f}{g}[\/latex].<\/p>\n<p id=\"fs-id1165042988256\">\u25a1<\/p>\n<div id=\"fs-id1165043397578\" class=\"textbox examples\">\n<h3>Applying L\u2019H\u00f4pital\u2019s Rule (0\/0 Case)<\/h3>\n<div id=\"fs-id1165043104016\" class=\"exercise\">\n<div id=\"fs-id1165043395456\" class=\"textbox\">\n<p id=\"fs-id1165042456913\">Evaluate each of the following limits by applying L\u2019H\u00f4pital\u2019s rule.<\/p>\n<ol style=\"list-style-type: lower-alpha\">\n<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{1- \\cos x}{x}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 1}{\\lim}\\frac{\\sin (\\pi x)}{\\ln x}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^{1\/x}-1}{1\/x}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042535045\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042535045\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165042535045\" style=\"list-style-type: lower-alpha\">\n<li>Since the numerator [latex]1- \\cos x\\to 0[\/latex] and the denominator [latex]x\\to 0[\/latex], we can apply L\u2019H\u00f4pital\u2019s rule to evaluate this limit. We have\n<div id=\"fs-id1165042328696\" class=\"equation unnumbered\">[latex]\\begin{array}{ll} \\underset{x\\to 0}{\\lim}\\frac{1- \\cos x}{x} & =\\underset{x\\to 0}{\\lim}\\frac{\\frac{d}{dx}(1- \\cos x)}{\\frac{d}{dx}(x)} \\\\ & =\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{1} \\\\ & =\\frac{\\underset{x\\to 0}{\\lim}(\\sin x)}{\\underset{x\\to 0}{\\lim}(1)} \\\\ & =\\frac{0}{1}=0 \\end{array}[\/latex]<\/div>\n<\/li>\n<li>As [latex]x\\to 1[\/latex], the numerator [latex]\\sin (\\pi x)\\to 0[\/latex] and the denominator [latex]\\ln x \\to 0[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule. We obtain\n<div id=\"fs-id1165043116372\" class=\"equation unnumbered\">[latex]\\begin{array}{ll} \\underset{x\\to 1}{\\lim}\\frac{\\sin (\\pi x)}{\\ln x} & =\\underset{x\\to 1}{\\lim}\\frac{\\pi \\cos (\\pi x)}{1\/x} \\\\ & =\\underset{x\\to 1}{\\lim}(\\pi x) \\cos (\\pi x) \\\\ & =(\\pi \\cdot 1)(-1)=\u2212\\pi \\end{array}[\/latex]<\/div>\n<\/li>\n<li>As [latex]x\\to \\infty[\/latex], the numerator [latex]e^{1\/x}-1\\to 0[\/latex] and the denominator [latex](\\frac{1}{x})\\to 0[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule. We obtain\n<div id=\"fs-id1165042327460\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^{1\/x}-1}{\\frac{1}{x}}=\\underset{x\\to \\infty }{\\lim}\\frac{e^{1\/x}(\\frac{-1}{x^2})}{(\\frac{-1}{x^2})}=\\underset{x\\to \\infty}{\\lim} e^{1\/x}=e^0=1[\/latex]<\/div>\n<\/li>\n<li>As [latex]x\\to 0[\/latex], both the numerator and denominator approach zero. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule. We obtain\n<div id=\"fs-id1165042562974\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}=\\underset{x\\to 0}{\\lim}\\frac{\\cos x-1}{2x}[\/latex].<\/div>\n<p>Since the numerator and denominator of this new quotient both approach zero as [latex]x\\to 0[\/latex], we apply L\u2019H\u00f4pital\u2019s rule again. In doing so, we see that<\/p>\n<div id=\"fs-id1165043281524\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\cos x-1}{2x}=\\underset{x\\to 0}{\\lim}\\frac{\u2212\\sin x}{2}=0[\/latex].<\/div>\n<p>Therefore, we conclude that<\/p>\n<div id=\"fs-id1165043284142\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}=0[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043393593\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165043395209\" class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1165043395214\">Evaluate [latex]\\underset{x\\to 0}{\\lim}\\frac{x}{\\tan x}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042377480\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042377480\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042377480\">1<\/p>\n<\/div>\n<div id=\"fs-id1165043395671\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043395678\">[latex]\\frac{d}{dx} \\tan x= \\sec ^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042318672\">We can also use L\u2019H\u00f4pital\u2019s rule to evaluate limits of quotients [latex]\\frac{f(x)}{g(x)}[\/latex] in which [latex]f(x)\\to \\pm \\infty[\/latex] and [latex]g(x)\\to \\pm \\infty[\/latex]. Limits of this form are classified as <em>indeterminate forms of type<\/em> [latex]\\infty \/ \\infty[\/latex]. Again, note that we are not actually dividing [latex]\\infty[\/latex] by [latex]\\infty[\/latex]. Since [latex]\\infty[\/latex] is not a real number, that is impossible; rather, [latex]\\infty \/ \\infty[\/latex] is used to represent a quotient of limits, each of which is [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex].<\/p>\n<div id=\"fs-id1165042330826\" class=\"textbox key-takeaways theorem\">\n<h3>L\u2019H\u00f4pital\u2019s Rule ([latex]\\infty \/ \\infty[\/latex] Case)<\/h3>\n<p id=\"fs-id1165043426174\">Suppose [latex]f[\/latex] and [latex]g[\/latex] are differentiable functions over an open interval containing [latex]a[\/latex], except possibly at [latex]a[\/latex]. Suppose [latex]\\underset{x\\to a}{\\lim}f(x)=\\infty[\/latex] (or [latex]\u2212\\infty[\/latex]) and [latex]\\underset{x\\to a}{\\lim}g(x)=\\infty[\/latex] (or [latex]\u2212\\infty[\/latex]). Then,<\/p>\n<div id=\"fs-id1165042373703\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex],<\/div>\n<p id=\"fs-id1165042707280\">assuming the limit on the right exists or is [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex]. This result also holds if the limit is infinite, if [latex]a=\\infty[\/latex] or [latex]\u2212\\infty[\/latex], or the limit is one-sided.<\/p>\n<\/div>\n<div id=\"fs-id1165043427623\" class=\"textbox examples\">\n<h3>Applying L\u2019H\u00f4pital\u2019s Rule ([latex]\\infty \/\\infty[\/latex] Case)<\/h3>\n<div id=\"fs-id1165043427625\" class=\"exercise\">\n<div id=\"fs-id1165042323490\" class=\"textbox\">\n<p id=\"fs-id1165042709794\">Evaluate each of the following limits by applying L\u2019H\u00f4pital\u2019s rule.<\/p>\n<ol id=\"fs-id1165042709798\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\underset{x\\to \\infty }{\\lim}\\frac{3x+5}{2x+1}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{\\cot x}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042376758\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042376758\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165042376758\" style=\"list-style-type: lower-alpha\">\n<li>Since [latex]3x+5[\/latex] and [latex]2x+1[\/latex] are first-degree polynomials with positive leading coefficients, [latex]\\underset{x\\to \\infty }{\\lim}(3x+5)=\\infty[\/latex] and [latex]\\underset{x\\to \\infty }{\\lim}(2x+1)=\\infty[\/latex]. Therefore, we apply L\u2019H\u00f4pital\u2019s rule and obtain\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty}{\\lim}\\frac{3x+5}{2x+1}=\\underset{x\\to \\infty }{\\lim}\\frac{3}{2}=\\frac{3}{2}[\/latex].<\/div>\n<p>Note that this limit can also be calculated without invoking L\u2019H\u00f4pital\u2019s rule. Earlier in the chapter we showed how to evaluate such a limit by dividing the numerator and denominator by the highest power of [latex]x[\/latex] in the denominator. In doing so, we saw that<\/p>\n<div id=\"fs-id1165042319986\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{3x+5}{2x+1}=\\underset{x\\to \\infty }{\\lim}\\frac{3+5\/x}{2+1\/x}=\\frac{3}{2}[\/latex].<\/div>\n<p>L\u2019H\u00f4pital\u2019s rule provides us with an alternative means of evaluating this type of limit.<\/li>\n<li>Here, [latex]\\underset{x\\to 0^+}{\\lim} \\ln x=\u2212\\infty[\/latex] and [latex]\\underset{x\\to 0^+}{\\lim} \\cot x=\\infty[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule and obtain\n<div id=\"fs-id1165042374803\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{\\cot x}=\\underset{x\\to 0^+}{\\lim}\\frac{1\/x}{\u2212\\csc^2 x}=\\underset{x\\to 0^+}{\\lim}\\frac{1}{\u2212x \\csc^2 x}[\/latex].<\/div>\n<p>Now as [latex]x\\to 0^+[\/latex], [latex]\\csc^2 x\\to \\infty[\/latex]. Therefore, the first term in the denominator is approaching zero and the second term is getting really large. In such a case, anything can happen with the product. Therefore, we cannot make any conclusion yet. To evaluate the limit, we use the definition of [latex]\\csc x[\/latex] to write<\/p>\n<div id=\"fs-id1165042376466\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{\u2212x \\csc^2 x}=\\underset{x\\to 0^+}{\\lim}\\frac{\\sin^2 x}{\u2212x}[\/latex].<\/div>\n<p>Now [latex]\\underset{x\\to 0^+}{\\lim} \\sin^2 x=0[\/latex] and [latex]\\underset{x\\to 0^+}{\\lim} x=0[\/latex], so we apply L\u2019H\u00f4pital\u2019s rule again. We find<\/p>\n<div id=\"fs-id1165043249678\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\sin^2 x}{\u2212x}=\\underset{x\\to 0^+}{\\lim}\\frac{2 \\sin x \\cos x}{-1}=\\frac{0}{-1}=0[\/latex].<\/div>\n<p>We conclude that<\/p>\n<div id=\"fs-id1165042315721\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{\\cot x}=0[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042333392\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042708180\" class=\"exercise\">\n<div id=\"fs-id1165042708182\" class=\"textbox\">\n<p id=\"fs-id1165042708184\">Evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{5x}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165042367881\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042367881\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042367881\" class=\"hidden-answer\" style=\"display: none\">0<\/div>\n<div id=\"fs-id1165042367890\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042705514\">[latex]\\frac{d}{dx}\\ln x=\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042320327\">As mentioned, L\u2019H\u00f4pital\u2019s rule is an extremely useful tool for evaluating limits. It is important to remember, however, that to apply L\u2019H\u00f4pital\u2019s rule to a quotient [latex]\\frac{f(x)}{g(x)}[\/latex], it is essential that the limit of [latex]\\frac{f(x)}{g(x)}[\/latex] be of the form [latex]0\/0[\/latex] or [latex]\\infty \/ \\infty[\/latex] Consider the following example.<\/p>\n<div id=\"fs-id1165042350228\" class=\"textbox examples\">\n<h3>When L\u2019H\u00f4pital\u2019s Rule Does Not Apply<\/h3>\n<div id=\"fs-id1165043219076\" class=\"exercise\">\n<div id=\"fs-id1165043219079\" class=\"textbox\">\n<p id=\"fs-id1165043219084\">Consider [latex]\\underset{x\\to 1}{\\lim}\\frac{x^2+5}{3x+4}[\/latex]. Show that the limit cannot be evaluated by applying L\u2019H\u00f4pital\u2019s rule.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042327372\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042327372\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042327372\">Because the limits of the numerator and denominator are not both zero and are not both infinite, we cannot apply L\u2019H\u00f4pital\u2019s rule. If we try to do so, we get<\/p>\n<div id=\"fs-id1165042398961\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(x^2+5)=2x[\/latex]<\/div>\n<p id=\"fs-id1165043395079\">and<\/p>\n<div id=\"fs-id1165042318693\" class=\"equation unnumbered\">[latex]\\frac{d}{dx}(3x+4)=3[\/latex].<\/div>\n<p id=\"fs-id1165042364614\">At which point we would conclude erroneously that<\/p>\n<div id=\"fs-id1165042364618\" class=\"equation unnumbered\">[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2+5}{3x+4}=\\underset{x\\to 1}{\\lim}\\frac{2x}{3}=\\frac{2}{3}[\/latex].<\/div>\n<p id=\"fs-id1165043222028\">However, since [latex]\\underset{x\\to 1}{\\lim}(x^2+5)=6[\/latex] and [latex]\\underset{x\\to 1}{\\lim}(3x+4)=7[\/latex], we actually have<\/p>\n<div id=\"fs-id1165042970462\" class=\"equation unnumbered\">[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2+5}{3x+4}=\\frac{6}{7}[\/latex].<\/div>\n<p id=\"fs-id1165042383150\">We can conclude that<\/p>\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2+5}{3x+4}\\ne \\underset{x\\to 1}{\\lim}\\frac{\\frac{d}{dx}(x^2+5)}{\\frac{d}{dx}(3x+4)}[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043174646\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165043174649\" class=\"exercise\">\n<div id=\"fs-id1165043174651\" class=\"textbox\">\n<p id=\"fs-id1165043174654\">Explain why we cannot apply L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\cos x}{x}[\/latex]. Evaluate [latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\cos x}{x}[\/latex] by other means.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042632518\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042632518\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042632518\">[latex]\\underset{x\\to 0^+}{\\lim} \\cos x=1[\/latex]. Therefore, we cannot apply L\u2019H\u00f4pital\u2019s rule. The limit of the quotient is [latex]\\infty[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042632611\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042632618\">Determine the limits of the numerator and denominator separately.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042331793\" class=\"bc-section section\">\n<h1>Other Indeterminate Forms<\/h1>\n<p id=\"fs-id1165042331798\">L\u2019H\u00f4pital\u2019s rule is very useful for evaluating limits involving the indeterminate forms [latex]0\/0[\/latex] and [latex]\\infty \/ \\infty[\/latex]. However, we can also use L\u2019H\u00f4pital\u2019s rule to help evaluate limits involving other indeterminate forms that arise when evaluating limits. The expressions [latex]0 \\cdot \\infty[\/latex], [latex]\\infty - \\infty[\/latex], [latex]1^{\\infty}[\/latex], [latex]\\infty^0[\/latex], and [latex]0^0[\/latex] are all considered indeterminate forms. These expressions are not real numbers. Rather, they represent forms that arise when trying to evaluate certain limits. Next we realize why these are indeterminate forms and then understand how to use L\u2019H\u00f4pital\u2019s rule in these cases. The key idea is that we must rewrite the indeterminate forms in such a way that we arrive at the indeterminate form [latex]0\/0[\/latex] or [latex]\\infty \/ \\infty[\/latex].<\/p>\n<div id=\"fs-id1165042320288\" class=\"bc-section section\">\n<h2>Indeterminate Form of Type [latex]0 \\cdot \\infty[\/latex]<\/h2>\n<p id=\"fs-id1165042320301\">Suppose we want to evaluate [latex]\\underset{x\\to a}{\\lim}(f(x) \\cdot g(x))[\/latex], where [latex]f(x)\\to 0[\/latex] and [latex]g(x)\\to \\infty[\/latex] (or [latex]\u2212\\infty[\/latex]) as [latex]x\\to a[\/latex]. Since one term in the product is approaching zero but the other term is becoming arbitrarily large (in magnitude), anything can happen to the product. We use the notation [latex]0 \\cdot \\infty[\/latex] to denote the form that arises in this situation. The expression [latex]0 \\cdot \\infty[\/latex] is considered indeterminate because we cannot determine without further analysis the exact behavior of the product [latex]f(x)g(x)[\/latex] as [latex]x\\to \\infty[\/latex]. For example, let [latex]n[\/latex] be a positive integer and consider<\/p>\n<div id=\"fs-id1165043259749\" class=\"equation unnumbered\">[latex]f(x)=\\frac{1}{(x^n+1)}[\/latex] and [latex]g(x)=3x^2[\/latex].<\/div>\n<p id=\"fs-id1165042323522\">As [latex]x\\to \\infty[\/latex], [latex]f(x)\\to 0[\/latex] and [latex]g(x)\\to \\infty[\/latex]. However, the limit as [latex]x\\to \\infty[\/latex] of [latex]f(x)g(x)=\\frac{3x^2}{(x^n+1)}[\/latex] varies, depending on [latex]n[\/latex]. If [latex]n=2[\/latex], then [latex]\\underset{x\\to \\infty }{\\lim}f(x)g(x)=3[\/latex]. If [latex]n=1[\/latex], then [latex]\\underset{x\\to \\infty }{\\lim}f(x)g(x)=\\infty[\/latex]. If [latex]n=3[\/latex], then [latex]\\underset{x\\to \\infty }{\\lim}f(x)g(x)=0[\/latex]. Here we consider another limit involving the indeterminate form [latex]0 \\cdot \\infty[\/latex] and show how to rewrite the function as a quotient to use L\u2019H\u00f4pital\u2019s rule.<\/p>\n<div id=\"fs-id1165042368495\" class=\"textbox examples\">\n<h3>Indeterminate Form of Type [latex]0\u00b7\\infty[\/latex]<\/h3>\n<div id=\"fs-id1165042368497\" class=\"exercise\">\n<div id=\"fs-id1165042323649\" class=\"textbox\">\n<p id=\"fs-id1165042323663\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim}x \\ln x[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042545829\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042545829\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042545829\">First, rewrite the function [latex]x \\ln x[\/latex] as a quotient to apply L\u2019H\u00f4pital\u2019s rule. If we write<\/p>\n<div class=\"equation unnumbered\">[latex]x \\ln x=\\frac{\\ln x}{1\/x}[\/latex],<\/div>\n<p id=\"fs-id1165042383922\">we see that [latex]\\ln x\\to \u2212\\infty[\/latex] as [latex]x\\to 0^+[\/latex] and [latex]\\frac{1}{x}\\to \\infty[\/latex] as [latex]x\\to 0^+[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule and obtain<\/p>\n<div id=\"fs-id1165042705715\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{1\/x}=\\underset{x\\to 0^+}{\\lim}\\frac{\\frac{d}{dx}(\\ln x)}{\\frac{d}{dx}(1\/x)}=\\underset{x\\to 0^+}{\\lim}\\frac{1\/x}{-1\/x^2}=\\underset{x\\to 0^+}{\\lim}(\u2212x)=0[\/latex].<\/div>\n<p id=\"fs-id1165042318647\">We conclude that<\/p>\n<div id=\"fs-id1165042318650\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}x \\ln x=0[\/latex].<\/div>\n<div id=\"CNX_Calc_Figure_04_08_002\" class=\"wp-caption aligncenter\">\n<div style=\"width: 368px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211307\/CNX_Calc_Figure_04_08_004.jpg\" alt=\"The function y = x ln(x) is graphed for values x \u2265 0. At x = 0, the value of the function is 0.\" width=\"358\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2.<\/strong> Finding the limit at [latex]x=0[\/latex] of the function [latex]f(x)=x \\ln x[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043430905\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165043430908\" class=\"exercise\">\n<div id=\"fs-id1165043430910\" class=\"textbox\">\n<p id=\"fs-id1165043430912\">Evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cot x[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043286669\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043286669\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043286669\">1<\/p>\n<\/div>\n<div id=\"fs-id1165043286676\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043286682\">Write [latex]x \\cot x=\\frac{x \\cos x}{\\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042318802\" class=\"bc-section section\">\n<h2>Indeterminate Form of Type [latex]\\infty -\\infty[\/latex]<\/h2>\n<p id=\"fs-id1165042318816\">Another type of indeterminate form is [latex]\\infty -\\infty[\/latex]. Consider the following example. Let [latex]n[\/latex] be a positive integer and let [latex]f(x)=3x^n[\/latex] and [latex]g(x)=3x^2+5[\/latex]. As [latex]x\\to \\infty[\/latex], [latex]f(x)\\to \\infty[\/latex] and [latex]g(x)\\to \\infty[\/latex]. We are interested in [latex]\\underset{x\\to \\infty}{\\lim}(f(x)-g(x))[\/latex]. Depending on whether [latex]f(x)[\/latex] grows faster, [latex]g(x)[\/latex] grows faster, or they grow at the same rate, as we see next, anything can happen in this limit. Since [latex]f(x)\\to \\infty[\/latex] and [latex]g(x)\\to \\infty[\/latex], we write [latex]\\infty -\\infty[\/latex] to denote the form of this limit. As with our other indeterminate forms, [latex]\\infty -\\infty[\/latex] has no meaning on its own and we must do more analysis to determine the value of the limit. For example, suppose the exponent [latex]n[\/latex] in the function [latex]f(x)=3x^n[\/latex] is [latex]n=3[\/latex], then<\/p>\n<div id=\"fs-id1165043430807\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}(f(x)-g(x))=\\underset{x\\to \\infty }{\\lim}(3x^3-3x^2-5)=\\infty[\/latex].<\/div>\n<p id=\"fs-id1165042333220\">On the other hand, if [latex]n=2[\/latex], then<\/p>\n<div id=\"fs-id1165042333235\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}(f(x)-g(x))=\\underset{x\\to \\infty }{\\lim}(3x^2-3x^2-5)=-5[\/latex].<\/div>\n<p id=\"fs-id1165043395183\">However, if [latex]n=1[\/latex], then<\/p>\n<div id=\"fs-id1165043254251\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}(f(x)-g(x))=\\underset{x\\to \\infty }{\\lim}(3x-3x^2-5)=\u2212\\infty[\/latex].<\/div>\n<p id=\"fs-id1165043323851\">Therefore, the limit cannot be determined by considering only [latex]\\infty -\\infty[\/latex]. Next we see how to rewrite an expression involving the indeterminate form [latex]\\infty -\\infty[\/latex] as a fraction to apply L\u2019H\u00f4pital\u2019s rule.<\/p>\n<div id=\"fs-id1165043323875\" class=\"textbox examples\">\n<h3>Indeterminate Form of Type [latex]\\infty -\\infty[\/latex]<\/h3>\n<div id=\"fs-id1165043323877\" class=\"exercise\">\n<div id=\"fs-id1165043323879\" class=\"textbox\">\n<p id=\"fs-id1165042375663\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim}(\\frac{1}{x^2}-\\frac{1}{\\tan x})[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043281296\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043281296\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043281296\">By combining the fractions, we can write the function as a quotient. Since the least common denominator is [latex]x^2 \\tan x[\/latex], we have<\/p>\n<div id=\"fs-id1165043281316\" class=\"equation unnumbered\">[latex]\\frac{1}{x^2}-\\frac{1}{\\tan x}=\\frac{(\\tan x)-x^2}{x^2 \\tan x}[\/latex].<\/div>\n<p id=\"fs-id1165043259808\">As [latex]x\\to 0^+[\/latex], the numerator [latex]\\tan x-x^2 \\to 0[\/latex] and the denominator [latex]x^2 \\tan x \\to 0[\/latex]. Therefore, we can apply L\u2019H\u00f4pital\u2019s rule. Taking the derivatives of the numerator and the denominator, we have<\/p>\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{(\\tan x)-x^2}{x^2 \\tan x}=\\underset{x\\to 0^+}{\\lim}\\frac{(\\sec^2 x)-2x}{x^2 \\sec^2 x+2x \\tan x}[\/latex].<\/div>\n<p id=\"fs-id1165043327626\">As [latex]x\\to 0^+[\/latex], [latex](\\sec^2 x)-2x \\to 1[\/latex] and [latex]x^2 \\sec^2 x+2x \\tan x \\to 0[\/latex]. Since the denominator is positive as [latex]x[\/latex] approaches zero from the right, we conclude that<\/p>\n<div id=\"fs-id1165042710940\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{(\\sec^2 x)-2x}{x^2 \\sec^2 x+2x \\tan x}=\\infty[\/latex].<\/div>\n<p id=\"fs-id1165043396304\">Therefore,<\/p>\n<div id=\"fs-id1165043396307\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}(\\frac{1}{x^2}-\\frac{1}{ tan x})=\\infty[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043348549\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165043348552\" class=\"exercise\">\n<div id=\"fs-id1165043348554\" class=\"textbox\">\n<p id=\"fs-id1165043348557\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim}(\\frac{1}{x}-\\frac{1}{\\sin x})[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043317356\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043317356\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043317356\">0<\/p>\n<\/div>\n<div id=\"fs-id1165043317362\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043317368\">Rewrite the difference of fractions as a single fraction.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042364139\">Another type of indeterminate form that arises when evaluating limits involves exponents. The expressions [latex]0^0[\/latex], [latex]\\infty^0[\/latex], and [latex]1^{\\infty}[\/latex] are all indeterminate forms. On their own, these expressions are meaningless because we cannot actually evaluate these expressions as we would evaluate an expression involving real numbers. Rather, these expressions represent forms that arise when finding limits. Now we examine how L\u2019H\u00f4pital\u2019s rule can be used to evaluate limits involving these indeterminate forms.<\/p>\n<p id=\"fs-id1165042364178\">Since L\u2019H\u00f4pital\u2019s rule applies to quotients, we use the natural logarithm function and its properties to reduce a problem evaluating a limit involving exponents to a related problem involving a limit of a quotient. For example, suppose we want to evaluate [latex]\\underset{x\\to a}{\\lim}f(x)^{g(x)}[\/latex] and we arrive at the indeterminate form [latex]\\infty^0[\/latex]. (The indeterminate forms [latex]0^0[\/latex] and [latex]1^{\\infty}[\/latex] can be handled similarly.) We proceed as follows. Let<\/p>\n<div id=\"fs-id1165042709633\" class=\"equation unnumbered\">[latex]y=f(x)^{g(x)}[\/latex].<\/div>\n<p id=\"fs-id1165043250963\">Then,<\/p>\n<div id=\"fs-id1165043250966\" class=\"equation unnumbered\">[latex]\\ln y=\\ln (f(x)^{g(x)})=g(x) \\ln (f(x))[\/latex].<\/div>\n<p id=\"fs-id1165042640888\">Therefore,<\/p>\n<div id=\"fs-id1165042640891\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}[\\ln y]=\\underset{x\\to a}{\\lim}[g(x) \\ln (f(x))][\/latex].<\/div>\n<p id=\"fs-id1165043393684\">Since [latex]\\underset{x\\to a}{\\lim}f(x)=\\infty[\/latex], we know that [latex]\\underset{x\\to a}{\\lim}\\ln (f(x))=\\infty[\/latex]. Therefore, [latex]\\underset{x\\to a}{\\lim}g(x) \\ln (f(x))[\/latex] is of the indeterminate form [latex]0 \\cdot \\infty[\/latex], and we can use the techniques discussed earlier to rewrite the expression [latex]g(x) \\ln (f(x))[\/latex] in a form so that we can apply L\u2019H\u00f4pital\u2019s rule. Suppose [latex]\\underset{x\\to a}{\\lim}g(x) \\ln (f(x))=L[\/latex], where [latex]L[\/latex] may be [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex]. Then<\/p>\n<div id=\"fs-id1165042638509\" class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim}\\ln y=L[\/latex].<\/div>\n<p id=\"fs-id1165042638551\">Since the natural logarithm function is continuous, we conclude that<\/p>\n<div id=\"fs-id1165042638554\" class=\"equation unnumbered\">[latex]\\ln (\\underset{x\\to a}{\\lim} y)=L[\/latex]<\/div>\n<p id=\"fs-id1165042708323\">which gives us<\/p>\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to a}{\\lim} y=\\underset{x\\to a}{\\lim}f(x)^{g(x)}=e^L[\/latex].<\/div>\n<div id=\"fs-id1165043390815\" class=\"textbox examples\">\n<h3>Indeterminate Form of Type [latex]\\infty^0[\/latex]<\/h3>\n<div id=\"fs-id1165043390817\" class=\"exercise\">\n<div id=\"fs-id1165043390819\" class=\"textbox\">\n<p id=\"fs-id1165043390832\">Evaluate [latex]\\underset{x\\to \\infty }{\\lim} x^{1\/x}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043390866\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043390866\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043390866\">Let [latex]y=x^{1\/x}[\/latex]. Then,<\/p>\n<div id=\"fs-id1165043281565\" class=\"equation unnumbered\">[latex]\\ln (x^{1\/x})=\\frac{1}{x} \\ln x=\\frac{\\ln x}{x}[\/latex].<\/div>\n<p id=\"fs-id1165043281615\">We need to evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x}[\/latex]. Applying L\u2019H\u00f4pital\u2019s rule, we obtain<\/p>\n<div id=\"fs-id1165043281645\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim} \\ln y=\\underset{x\\to \\infty}{\\lim}\\frac{\\ln x}{x}=\\underset{x\\to \\infty}{\\lim}\\frac{1\/x}{1}=0[\/latex].<\/div>\n<p id=\"fs-id1165043173746\">Therefore, [latex]\\underset{x\\to \\infty }{\\lim}\\ln y=0[\/latex]. Since the natural logarithm function is continuous, we conclude that<\/p>\n<div class=\"equation unnumbered\">[latex]\\ln (\\underset{x\\to \\infty}{\\lim} y)=0[\/latex],<\/div>\n<p id=\"fs-id1165043427387\">which leads to<\/p>\n<div id=\"fs-id1165043427390\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim} y=\\underset{x\\to \\infty}{\\lim}\\frac{\\ln x}{x}=e^0=1[\/latex].<\/div>\n<p id=\"fs-id1165042407320\">Hence,<\/p>\n<div id=\"fs-id1165042407323\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty}{\\lim} x^{1\/x}=1[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042407362\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042407365\" class=\"exercise\">\n<div id=\"fs-id1165042407368\" class=\"textbox\">\n<p id=\"fs-id1165042407370\">Evaluate [latex]\\underset{x\\to \\infty}{\\lim} x^{1\/ \\ln x}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043108248\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043108248\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043108248\">[latex]e[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043108254\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165043108260\">Let [latex]y=x^{1\/ \\ln x}[\/latex] and apply the natural logarithm to both sides of the equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043108292\" class=\"textbox examples\">\n<h3>Indeterminate Form of Type [latex]0^0[\/latex]<\/h3>\n<div id=\"fs-id1165043108295\" class=\"exercise\">\n<div id=\"fs-id1165043108297\" class=\"textbox\">\n<p id=\"fs-id1165042657720\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim} x^{\\sin x}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042657755\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042657755\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042657755\">Let<\/p>\n<div id=\"fs-id1165042657759\" class=\"equation unnumbered\">[latex]y=x^{\\sin x}[\/latex].<\/div>\n<p id=\"fs-id1165042657780\">Therefore,<\/p>\n<div id=\"fs-id1165042657783\" class=\"equation unnumbered\">[latex]\\ln y=\\ln (x^{\\sin x})= \\sin x \\ln x[\/latex].<\/div>\n<p id=\"fs-id1165042707171\">We now evaluate [latex]\\underset{x\\to 0^+}{\\lim} \\sin x \\ln x[\/latex]. Since [latex]\\underset{x\\to 0^+}{\\lim} \\sin x=0[\/latex] and [latex]\\underset{x\\to 0^+}{\\lim} \\ln x=\u2212\\infty[\/latex], we have the indeterminate form [latex]0 \\cdot \\infty[\/latex]. To apply L\u2019H\u00f4pital\u2019s rule, we need to rewrite [latex]\\sin x \\ln x[\/latex] as a fraction. We could write<\/p>\n<div id=\"fs-id1165043173832\" class=\"equation unnumbered\">[latex]\\sin x \\ln x=\\frac{\\sin x}{1\/ \\ln x}[\/latex]<\/div>\n<p id=\"fs-id1165043173865\">or<\/p>\n<div class=\"equation unnumbered\">[latex]\\sin x \\ln x=\\frac{\\ln x}{1\/ \\sin x}=\\frac{\\ln x}{\\csc x}[\/latex].<\/div>\n<p id=\"fs-id1165042364489\">Let\u2019s consider the first option. In this case, applying L\u2019H\u00f4pital\u2019s rule, we would obtain<\/p>\n<div id=\"fs-id1165042364494\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim} \\sin x \\ln x=\\underset{x\\to 0^+}{\\lim}\\frac{\\sin x}{1\/ \\ln x}=\\underset{x\\to 0^+}{\\lim}\\frac{\\cos x}{-1\/(x(\\ln x)^2)}=\\underset{x\\to 0^+}{\\lim}(\u2212x(\\ln x)^2 \\cos x)[\/latex].<\/div>\n<p id=\"fs-id1165043317274\">Unfortunately, we not only have another expression involving the indeterminate form [latex]0 \\cdot \\infty[\/latex], but the new limit is even more complicated to evaluate than the one with which we started. Instead, we try the second option. By writing<\/p>\n<div id=\"fs-id1165043131555\" class=\"equation unnumbered\">[latex]\\sin x \\ln x=\\frac{\\ln x}{1\/ \\sin x}=\\frac{\\ln x}{\\csc x}[\/latex]<\/div>\n<p id=\"fs-id1165043131604\">and applying L\u2019H\u00f4pital\u2019s rule, we obtain<\/p>\n<div id=\"fs-id1165043131609\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim} \\sin x \\ln x=\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{\\csc x}=\\underset{x\\to 0^+}{\\lim}\\frac{1\/x}{\u2212 \\csc x \\cot x}=\\underset{x\\to 0^+}{\\lim}\\frac{-1}{x \\csc x \\cot x}[\/latex].<\/div>\n<p id=\"fs-id1165042651533\">Using the fact that [latex]\\csc x=\\frac{1}{\\sin x}[\/latex] and [latex]\\cot x=\\frac{\\cos x}{\\sin x}[\/latex], we can rewrite the expression on the right-hand side as<\/p>\n<div id=\"fs-id1165043251999\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\u2212\\sin^2 x}{x \\cos x}=\\underset{x\\to 0^+}{\\lim}[\\frac{\\sin x}{x} \\cdot (\u2212\\tan x)]=(\\underset{x\\to 0^+}{\\lim}\\frac{\\sin x}{x}) \\cdot (\\underset{x\\to 0^+}{\\lim}(\u2212\\tan x))=1 \\cdot 0=0[\/latex].<\/div>\n<p id=\"fs-id1165042676314\">We conclude that [latex]\\underset{x\\to 0^+}{\\lim} \\ln y=0[\/latex]. Therefore, [latex]\\ln (\\underset{x\\to 0^+}{\\lim} y)=0[\/latex] and we have<\/p>\n<div id=\"fs-id1165042327527\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim} y=\\underset{x\\to 0^+}{\\lim} x^{\\sin x}=e^0=1[\/latex].<\/div>\n<p id=\"fs-id1165042327592\">Hence,<\/p>\n<div id=\"fs-id1165042327595\" class=\"equation unnumbered\">[latex]\\underset{x\\to 0^+}{\\lim} x^{\\sin x}=1[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042660254\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042660257\" class=\"exercise\">\n<div id=\"fs-id1165042660259\" class=\"textbox\">\n<p id=\"fs-id1165042660261\">Evaluate [latex]\\underset{x\\to 0^+}{\\lim} x^x[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042660293\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042660293\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042660293\">1<\/p>\n<\/div>\n<div id=\"fs-id1165042660300\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042660306\">Let [latex]y=x^x[\/latex] and take the natural logarithm of both sides of the equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042660328\" class=\"bc-section section\">\n<h1>Growth Rates of Functions<\/h1>\n<p id=\"fs-id1165042660333\">Suppose the functions [latex]f[\/latex] and [latex]g[\/latex] both approach infinity as [latex]x\\to \\infty[\/latex]. Although the values of both functions become arbitrarily large as the values of [latex]x[\/latex] become sufficiently large, sometimes one function is growing more quickly than the other. For example, [latex]f(x)=x^2[\/latex] and [latex]g(x)=x^3[\/latex] both approach infinity as [latex]x\\to \\infty[\/latex]. However, as shown in the following table, the values of [latex]x^3[\/latex] are growing much faster than the values of [latex]x^2[\/latex].<\/p>\n<table id=\"fs-id1165042700402\" class=\"column-header\" summary=\"This table has three rows and five columns. The first column is a header column, and it reads from top to bottom x, f(x) = x2, and g(x) = x3. To the right of the header, the first row reads 10, 100, 1000, and 10,000. The second row reads 100, 10,000, 1,000,000, and 100,000,000. The third row reads 1000, 1,000,000, 1,000,000,000, and 1,000,000,000,000.\">\n<caption>Comparing the Growth Rates of [latex]x^2[\/latex] and [latex]x^3[\/latex]<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>10<\/td>\n<td>100<\/td>\n<td>1000<\/td>\n<td>10,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]f(x)=x^2[\/latex]<\/strong><\/td>\n<td>100<\/td>\n<td>10,000<\/td>\n<td>1,000,000<\/td>\n<td>100,000,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]g(x)=x^3[\/latex]<\/strong><\/td>\n<td>1000<\/td>\n<td>1,000,000<\/td>\n<td>1,000,000,000<\/td>\n<td>[latex]1,000,000,000,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165043219185\">In fact,<\/p>\n<div id=\"fs-id1165043219189\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{x^3}{x^2}=\\underset{x\\to \\infty}{\\lim} x=\\infty[\/latex] or, equivalently, [latex]\\underset{x\\to \\infty}{\\lim}\\frac{x^2}{x^3}=\\underset{x\\to \\infty }{\\lim}\\frac{1}{x}=0[\/latex].<\/div>\n<p id=\"fs-id1165042465546\">As a result, we say [latex]x^3[\/latex] is growing more rapidly than [latex]x^2[\/latex] as [latex]x\\to \\infty[\/latex]. On the other hand, for [latex]f(x)=x^2[\/latex] and [latex]g(x)=3x^2+4x+1[\/latex], although the values of [latex]g(x)[\/latex] are always greater than the values of [latex]f(x)[\/latex] for [latex]x>0[\/latex], each value of [latex]g(x)[\/latex] is roughly three times the corresponding value of [latex]f(x)[\/latex] as [latex]x\\to \\infty[\/latex], as shown in the following table. In fact,<\/p>\n<div id=\"fs-id1165042461121\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{x^2}{3x^2+4x+1}=\\frac{1}{3}[\/latex].<\/div>\n<table id=\"fs-id1165042461176\" class=\"column-header\" summary=\"This table has three rows and five columns. The first column is a header column, and it reads from top to bottom x, f(x) = x2, and g(x) = 3x2 + 4x + 1. To the right of the header, the first row reads 10, 100, 1000, and 10,000. The second row reads 100, 10,000, 1,000,000, and 100,000,000. The third row reads 341, 30,401, 3,004,001, and 300,040,001.\">\n<caption>Comparing the Growth Rates of [latex]x^2[\/latex] and [latex]3x^2+4x+1[\/latex]<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>10<\/td>\n<td>100<\/td>\n<td>1000<\/td>\n<td>10,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]f(x)=x^2[\/latex]<\/strong><\/td>\n<td>100<\/td>\n<td>10,000<\/td>\n<td>1,000,000<\/td>\n<td>100,000,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]g(x)=3x^2+4x+1[\/latex]<\/strong><\/td>\n<td>341<\/td>\n<td>30,401<\/td>\n<td>3,004,001<\/td>\n<td>300,040,001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042659418\">In this case, we say that [latex]x^2[\/latex] and [latex]3x^2+4x+1[\/latex] are growing at the same rate as [latex]x\\to \\infty[\/latex].<\/p>\n<p id=\"fs-id1165042659463\">More generally, suppose [latex]f[\/latex] and [latex]g[\/latex] are two functions that approach infinity as [latex]x\\to \\infty[\/latex]. We say [latex]g[\/latex] grows more rapidly than [latex]f[\/latex] as [latex]x\\to \\infty[\/latex] if<\/p>\n<div id=\"fs-id1165042659507\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{g(x)}{f(x)}=\\infty[\/latex] or, equivalently, [latex]\\underset{x\\to \\infty }{\\lim}\\frac{f(x)}{g(x)}=0[\/latex].<\/div>\n<p id=\"fs-id1165043422372\">On the other hand, if there exists a constant [latex]M \\ne 0[\/latex] such that<\/p>\n<div id=\"fs-id1165043422384\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{f(x)}{g(x)}=M[\/latex],<\/div>\n<p id=\"fs-id1165043422432\">we say [latex]f[\/latex] and [latex]g[\/latex] grow at the same rate as [latex]x\\to \\infty[\/latex].<\/p>\n<p id=\"fs-id1165043422456\">Next we see how to use L\u2019H\u00f4pital\u2019s rule to compare the growth rates of power, exponential, and logarithmic functions.<\/p>\n<div id=\"fs-id1165043422462\" class=\"textbox examples\">\n<h3>Comparing the Growth Rates of [latex]\\ln x[\/latex], [latex]x^2[\/latex], and [latex]e^x[\/latex]<\/h3>\n<div id=\"fs-id1165043422464\" class=\"exercise\">\n<div id=\"fs-id1165043422466\" class=\"textbox\">\n<p id=\"fs-id1165042325994\">For each of the following pairs of functions, use L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to \\infty }{\\lim}(\\frac{f(x)}{g(x)})[\/latex].<\/p>\n<ol id=\"fs-id1165042326044\" style=\"list-style-type: lower-alpha\">\n<li>[latex]f(x)=x^2[\/latex] and [latex]g(x)=e^x[\/latex]<\/li>\n<li>[latex]f(x)=\\ln x[\/latex] and [latex]g(x)=x^2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042326147\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042326147\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165042326147\" style=\"list-style-type: lower-alpha\">\n<li>Since [latex]\\underset{x\\to \\infty }{\\lim} x^2=\\infty[\/latex] and [latex]\\underset{x\\to \\infty }{\\lim} e^x[\/latex], we can use L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to \\infty }{\\lim}[\\frac{x^2}{e^x}][\/latex]. We obtain\n<div id=\"fs-id1165042631803\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{x^2}{e^x}=\\underset{x\\to \\infty }{\\lim}\\frac{2x}{e^x}[\/latex].<\/div>\n<p>Since [latex]\\underset{x\\to \\infty }{\\lim}2x=\\infty[\/latex] and [latex]\\underset{x\\to \\infty }{\\lim}e^x=\\infty[\/latex], we can apply L\u2019H\u00f4pital\u2019s rule again. Since<\/p>\n<div id=\"fs-id1165042631923\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{2x}{e^x}=\\underset{x\\to \\infty }{\\lim}\\frac{2}{e^x}=0[\/latex],<\/div>\n<p>we conclude that<\/p>\n<div class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{x^2}{e^x}=0[\/latex].<\/div>\n<p>Therefore, [latex]e^x[\/latex] grows more rapidly than [latex]x^2[\/latex] as [latex]x\\to \\infty[\/latex] (See <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_08_003\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165042418200\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_04_08_003\" class=\"wp-caption aligncenter\">\n<div style=\"width: 281px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211309\/CNX_Calc_Figure_04_08_005.jpg\" alt=\"The functions g(x) = ex and f(x) = x2 are graphed. It is obvious that g(x) increases much more quickly than f(x).\" width=\"271\" height=\"278\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3.<\/strong> An exponential function grows at a faster rate than a power function.<\/p>\n<\/div>\n<\/div>\n<table id=\"fs-id1165042418200\" class=\"column-header\" summary=\"This table has three rows and five columns. The first column is a header column, and it reads from top to bottom x, x2, and ex. To the right of the header, the first row reads 5, 10, 15, and 20. The second row reads 25, 100, 225, and 400. The third row reads 148, 22,026, 3,269,017, and 485,165,195.\">\n<caption>Growth rates of a power function and an exponential function.<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>5<\/td>\n<td>10<\/td>\n<td>15<\/td>\n<td>20<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x^2[\/latex]<\/strong><\/td>\n<td>25<\/td>\n<td>100<\/td>\n<td>225<\/td>\n<td>400<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]e^x[\/latex]<\/strong><\/td>\n<td>148<\/td>\n<td>22,026<\/td>\n<td>3,269,017<\/td>\n<td>485,165,195<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Since [latex]\\underset{x\\to \\infty }{\\lim} \\ln x=\\infty[\/latex] and [latex]\\underset{x\\to \\infty }{\\lim} x^2=\\infty[\/latex], we can use L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x^2}[\/latex]. We obtain\n<div id=\"fs-id1165042471196\" class=\"equation unnumbered\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x^2}=\\underset{x\\to \\infty }{\\lim}\\frac{1\/x}{2x}=\\underset{x\\to \\infty }{\\lim}\\frac{1}{2x^2}=0[\/latex].<\/div>\n<p>Thus, [latex]x^2[\/latex] grows more rapidly than [latex]\\ln x[\/latex] as [latex]x\\to \\infty[\/latex] (see <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_08_004\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165042471354\">(Figure)<\/a>).<\/p>\n<div id=\"CNX_Calc_Figure_04_08_004\" class=\"wp-caption aligncenter\">\n<div style=\"width: 427px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211312\/CNX_Calc_Figure_04_08_006.jpg\" alt=\"The functions g(x) = x2 and f(x) = ln(x) are graphed. It is obvious that g(x) increases much more quickly than f(x).\" width=\"417\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 4.<\/strong> A power function grows at a faster rate than a logarithmic function.<\/p>\n<\/div>\n<\/div>\n<table id=\"fs-id1165042471354\" class=\"column-header\" summary=\"This table has three rows and five columns. The first column is a header column, and it reads from top to bottom x, ln(x), and x2. To the right of the header, the first row reads 10, 100, 1000, and 10,000. The second row reads 2.303, 4.605, 6.908, and 9.210. The third row reads 100, 10,000, 1,000,000, and 100,000,000.\">\n<caption>Growth rates of a power function and a logarithmic function<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>10<\/td>\n<td>100<\/td>\n<td>1000<\/td>\n<td>10,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]\\ln x[\/latex]<\/strong><\/td>\n<td>2.303<\/td>\n<td>4.605<\/td>\n<td>6.908<\/td>\n<td>9.210<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x^2[\/latex]<\/strong><\/td>\n<td>100<\/td>\n<td>10,000<\/td>\n<td>1,000,000<\/td>\n<td>100,000,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042463715\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1165042463719\" class=\"exercise\">\n<div id=\"fs-id1165042463721\" class=\"textbox\">\n<p id=\"fs-id1165042463723\">Compare the growth rates of [latex]x^{100}[\/latex] and [latex]2^x[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042463749\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042463749\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042463749\">The function [latex]2^x[\/latex] grows faster than [latex]x^{100}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165042463772\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1165042463780\">Apply L\u2019H\u00f4pital\u2019s rule to [latex]x^{100}\/2^x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042463803\">Using the same ideas as in <a class=\"autogenerated-content\" href=\"#fs-id1165043422462\">(Figure)<\/a>a. it is not difficult to show that [latex]e^x[\/latex] grows more rapidly than [latex]x^p[\/latex] for any [latex]p>0[\/latex]. In <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_08_005\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165043327297\">(Figure)<\/a>, we compare [latex]e^x[\/latex] with [latex]x^3[\/latex] and [latex]x^4[\/latex] as [latex]x\\to \\infty[\/latex].<\/p>\n<div id=\"CNX_Calc_Figure_04_08_005\" class=\"wp-caption aligncenter\">\n<div style=\"width: 868px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211316\/CNX_Calc_Figure_04_08_001.jpg\" alt=\"This figure has two figures marked a and b. In figure a, the functions y = ex and y = x3 are graphed. It is obvious that ex increases more quickly than x3. In figure b, the functions y = ex and y = x4 are graphed. It is obvious that ex increases much more quickly than x4, but the point at which that happens is further to the right than it was for x3.\" width=\"858\" height=\"386\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 5.<\/strong> The exponential function [latex]e^x[\/latex] grows faster than [latex]x^p[\/latex] for any [latex]p&gt;0[\/latex]. (a) A comparison of [latex]e^x[\/latex] with [latex]x^3[\/latex]. (b) A comparison of [latex]e^x[\/latex] with [latex]x^4[\/latex].<\/p>\n<\/div>\n<\/div>\n<table class=\"column-header\" summary=\"This table has four rows and five columns. The first column is a header column, and it reads from top to bottom x, x3, x4, and ex. To the right of the header, the first row reads 5, 10, 15, and 20. The second row reads 125, 1000, 3375, and 8000. The third row reads 625, 10,000, 50,625, and 160,000. The fourth row reads 148, 22,026, 3,269,017, and 485,165,195.\">\n<caption>An exponential function grows at a faster rate than any power function<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>5<\/td>\n<td>10<\/td>\n<td>15<\/td>\n<td>20<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x^3[\/latex]<\/strong><\/td>\n<td>125<\/td>\n<td>1000<\/td>\n<td>3375<\/td>\n<td>8000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]x^4[\/latex]<\/strong><\/td>\n<td>625<\/td>\n<td>10,000<\/td>\n<td>50,625<\/td>\n<td>160,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]e^x[\/latex]<\/strong><\/td>\n<td>148<\/td>\n<td>22,026<\/td>\n<td>3,269,017<\/td>\n<td>485,165,195<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042542902\">Similarly, it is not difficult to show that [latex]x^p[\/latex] grows more rapidly than [latex]\\ln x[\/latex] for any [latex]p>0[\/latex]. In <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_04_08_006\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#fs-id1165042460279\">(Figure)<\/a>, we compare [latex]\\ln x[\/latex] with [latex]\\sqrt[3]{x}[\/latex] and [latex]\\sqrt{x}[\/latex].<\/p>\n<div id=\"CNX_Calc_Figure_04_08_006\" class=\"wp-caption aligncenter\">\n<div style=\"width: 353px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211319\/CNX_Calc_Figure_04_08_007.jpg\" alt=\"This figure shows y = the square root of x, y = the cube root of x, and y = ln(x). It is apparent that y = ln(x) grows more slowly than either of these functions.\" width=\"343\" height=\"203\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6.<\/strong> The function [latex]y=\\ln x[\/latex] grows more slowly than [latex]x^p[\/latex] for any [latex]p&gt;0[\/latex] as [latex]x\\to \\infty[\/latex].<\/p>\n<\/div>\n<\/div>\n<table id=\"fs-id1165042460279\" class=\"column-header\" summary=\"This table has four rows and five columns. The first column is a header column, and it reads from top to bottom x, ln(x), the cube root of x, and the square root of x. To the right of the header, the first row reads 10, 100, 1000, and 10,000. The second row reads 2.303, 4.605, 6.908, and 9.210. The third row reads 2.154, 4.642, 10, and 21.544. The fourth row reads 3.162, 10, 31.623, and 100.\">\n<caption>A logarithmic function grows at a slower rate than any root function<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>10<\/td>\n<td>100<\/td>\n<td>1000<\/td>\n<td>10,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]\\ln x[\/latex]<\/strong><\/td>\n<td>2.303<\/td>\n<td>4.605<\/td>\n<td>6.908<\/td>\n<td>9.210<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]\\sqrt[3]{x}[\/latex]<\/strong><\/td>\n<td>2.154<\/td>\n<td>4.642<\/td>\n<td>10<\/td>\n<td>21.544<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]\\sqrt{x}[\/latex]<\/strong><\/td>\n<td>3.162<\/td>\n<td>10<\/td>\n<td>31.623<\/td>\n<td>100<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165042658525\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165042658532\">\n<li>L\u2019H\u00f4pital\u2019s rule can be used to evaluate the limit of a quotient when the indeterminate form [latex]0\/0[\/latex] or [latex]\\infty \/ \\infty[\/latex] arises.<\/li>\n<li>L\u2019H\u00f4pital\u2019s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form [latex]0\/0[\/latex] or [latex]\\infty \/ \\infty[\/latex].<\/li>\n<li>The exponential function [latex]e^x[\/latex] grows faster than any power function [latex]x^p[\/latex], [latex]p>0[\/latex].<\/li>\n<li>The logarithmic function [latex]\\ln x[\/latex] grows more slowly than any power function [latex]x^p[\/latex], [latex]p>0[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165042658654\" class=\"textbox exercises\">\n<p id=\"fs-id1165043217918\">For the following exercises, evaluate the limit.<\/p>\n<div id=\"fs-id1165043217921\" class=\"exercise\">\n<div id=\"fs-id1165043217924\" class=\"textbox\">\n<p id=\"fs-id1165043217926\"><strong>1.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043217966\" class=\"exercise\">\n<div id=\"fs-id1165043217969\" class=\"textbox\">\n<p id=\"fs-id1165043217971\"><strong>2.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x^k}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043218009\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043218009\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043218009\">[latex]\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043218017\" class=\"exercise\">\n<div id=\"fs-id1165043218019\" class=\"textbox\">\n<p id=\"fs-id1165043218021\"><strong>3.<\/strong> Evaluate the limit [latex]\\underset{x\\to \\infty }{\\lim}\\frac{\\ln x}{x^k}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043218066\" class=\"exercise\">\n<div id=\"fs-id1165043218068\" class=\"textbox\">\n<p id=\"fs-id1165043218070\"><strong>4.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^2-a^2}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043218117\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043218117\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043218117\">[latex]\\frac{1}{2a}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043218132\" class=\"exercise\">\n<div id=\"fs-id1165043218134\" class=\"textbox\">\n<p id=\"fs-id1165043218136\"><strong>5.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^3-a^3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042469654\" class=\"exercise\">\n<div id=\"fs-id1165042469656\" class=\"textbox\">\n<p id=\"fs-id1165042469659\"><strong>6.<\/strong> Evaluate the limit [latex]\\underset{x\\to a}{\\lim}\\frac{x-a}{x^n-a^n}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042469706\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042469706\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042469706\">[latex]\\frac{1}{na^{n-1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042469729\">For the following exercises, determine whether you can apply L\u2019H\u00f4pital\u2019s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L\u2019H\u00f4pital\u2019s rule.<\/p>\n<div id=\"fs-id1165042469737\" class=\"exercise\">\n<div id=\"fs-id1165042469739\" class=\"textbox\">\n<p id=\"fs-id1165042469741\"><strong>7.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim}x^2 \\ln x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042469817\" class=\"exercise\">\n<div class=\"textbox\">\n<p><strong>8.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x^{1\/x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042711556\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042711556\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711556\">Cannot apply directly; use logarithms<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042711562\" class=\"exercise\">\n<div id=\"fs-id1165042711564\" class=\"textbox\">\n<p id=\"fs-id1165042711566\"><strong>9.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{2\/x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042711603\" class=\"exercise\">\n<div id=\"fs-id1165042711605\" class=\"textbox\">\n<p id=\"fs-id1165042711607\"><strong>10.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{x^2}{1\/x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042711643\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042711643\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711643\">Cannot apply directly; rewrite as [latex]\\underset{x\\to 0}{\\lim} x^3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042711670\" class=\"exercise\">\n<div id=\"fs-id1165042711672\" class=\"textbox\">\n<p id=\"fs-id1165042711674\"><strong>11.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}\\frac{e^x}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042711711\">For the following exercises, evaluate the limits with either L\u2019H\u00f4pital\u2019s rule or previously learned methods.<\/p>\n<div id=\"fs-id1165042711715\" class=\"exercise\">\n<div id=\"fs-id1165042711717\" class=\"textbox\">\n<p id=\"fs-id1165042711719\"><strong>12.<\/strong> [latex]\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x-3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042711760\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042711760\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711760\">6<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042711767\" class=\"exercise\">\n<div id=\"fs-id1165042711769\" class=\"textbox\">\n<p id=\"fs-id1165042711771\"><strong>13.<\/strong> [latex]\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x+3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042499499\" class=\"exercise\">\n<div id=\"fs-id1165042499502\" class=\"textbox\">\n<p id=\"fs-id1165042499504\"><strong>14.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^{-2}-1}{x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042499552\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042499552\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042499552\">-2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042499561\" class=\"exercise\">\n<div id=\"fs-id1165042499563\" class=\"textbox\">\n<p id=\"fs-id1165042499565\"><strong>15.<\/strong> [latex]\\underset{x\\to \\pi \/2}{\\lim}\\frac{\\cos x}{\\pi \/ 2 - x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042499616\" class=\"exercise\">\n<div id=\"fs-id1165042499618\" class=\"textbox\">\n<p id=\"fs-id1165042499620\"><strong>16.<\/strong> [latex]\\underset{x\\to \\pi }{\\lim}\\frac{x-\\pi }{\\sin x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042499655\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042499655\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042499655\">-1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042499664\" class=\"exercise\">\n<div id=\"fs-id1165042499666\" class=\"textbox\">\n<p id=\"fs-id1165042499668\"><strong>17.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{\\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042711252\" class=\"exercise\">\n<div id=\"fs-id1165042711254\" class=\"textbox\">\n<p id=\"fs-id1165042711256\"><strong>18.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^n-1}{x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042711303\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042711303\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711303\">[latex]n[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p><strong>19.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{(1+x)^n-1-nx}{x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042711396\" class=\"exercise\">\n<div id=\"fs-id1165042711398\" class=\"textbox\">\n<p id=\"fs-id1165042711401\"><strong>20.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x- \\tan x}{x^3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042711441\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042711441\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711441\">[latex]-\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1165042711459\"><strong>21.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sqrt{1+x}-\\sqrt{1-x}}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042602902\" class=\"exercise\">\n<div id=\"fs-id1165042602904\" class=\"textbox\">\n<p id=\"fs-id1165042602906\"><strong>22.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-x-1}{x^2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042602950\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042602950\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042602950\">[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042602961\" class=\"exercise\">\n<div id=\"fs-id1165042602964\" class=\"textbox\">\n<p id=\"fs-id1165042602966\"><strong>23.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\tan x}{\\sqrt{x}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042603005\" class=\"exercise\">\n<div id=\"fs-id1165042603007\" class=\"textbox\">\n<p id=\"fs-id1165042603009\"><strong>24.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{\\ln x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042603044\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042603044\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042603044\">1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042603052\" class=\"exercise\">\n<div id=\"fs-id1165042603054\" class=\"textbox\">\n<p id=\"fs-id1165042603056\"><strong>25.<\/strong> [latex]\\underset{x\\to 0}{\\lim}(x+1)^{1\/x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042603107\" class=\"exercise\">\n<div id=\"fs-id1165042603109\" class=\"textbox\">\n<p id=\"fs-id1165042603111\"><strong>26.<\/strong> [latex]\\underset{x\\to 1}{\\lim}\\frac{\\sqrt{x}-\\sqrt[3]{x}}{x-1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042617540\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042617540\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042617540\">[latex]\\frac{1}{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042617552\" class=\"exercise\">\n<div id=\"fs-id1165042617554\" class=\"textbox\">\n<p id=\"fs-id1165042617556\"><strong>27.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim} x^{2x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042617597\" class=\"exercise\">\n<div id=\"fs-id1165042617599\" class=\"textbox\">\n<p id=\"fs-id1165042617601\"><strong>28.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x \\sin (\\frac{1}{x})[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042617638\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042617638\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042617638\">1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042617646\" class=\"exercise\">\n<div id=\"fs-id1165042617648\" class=\"textbox\">\n<p id=\"fs-id1165042617650\"><strong>29.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x-x}{x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042617696\" class=\"exercise\">\n<div id=\"fs-id1165042617698\" class=\"textbox\">\n<p id=\"fs-id1165042617700\"><strong>30.<\/strong> [latex]\\underset{x\\to 0^+}{\\lim} x \\ln (x^4)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042617740\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042617740\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042617740\">0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042617748\" class=\"exercise\">\n<div id=\"fs-id1165042617750\" class=\"textbox\">\n<p id=\"fs-id1165042617752\"><strong>31.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim}(x-e^x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042707732\" class=\"exercise\">\n<div id=\"fs-id1165042707734\" class=\"textbox\">\n<p id=\"fs-id1165042707736\"><strong>32.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} x^2 e^{\u2212x}[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q849216\">Show Answer<\/span><\/p>\n<div id=\"q849216\" class=\"hidden-answer\" style=\"display: none\">0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042707778\" class=\"exercise\">\n<div id=\"fs-id1165042707780\" class=\"textbox\">\n<p id=\"fs-id1165042707782\"><strong>33.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{3^x-2^x}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042707843\" class=\"exercise\">\n<div id=\"fs-id1165042707845\" class=\"textbox\">\n<p><strong>34.<\/strong> [latex]\\underset{x\\to 0}{\\lim}\\frac{1+1\/x}{1-1\/x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042707893\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042707893\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042707893\">-1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1165042707904\" class=\"textbox\">\n<p id=\"fs-id1165042707906\"><strong>35.<\/strong> [latex]\\underset{x\\to \\pi \/4}{\\lim}(1- \\tan x) \\cot x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042707959\" class=\"exercise\">\n<div id=\"fs-id1165042707961\" class=\"textbox\">\n<p id=\"fs-id1165042707963\"><strong>36.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} xe^{1\/x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042525331\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042525331\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042525331\">[latex]\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042525338\" class=\"exercise\">\n<div id=\"fs-id1165042525340\" class=\"textbox\">\n<p id=\"fs-id1165042525342\"><strong>37.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{1\/ \\cos x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042525384\" class=\"exercise\">\n<div id=\"fs-id1165042525386\" class=\"textbox\">\n<p id=\"fs-id1165042525388\"><strong>38.<\/strong> [latex]\\underset{x\\to 0}{\\lim} x^{1\/x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042525418\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042525418\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042525418\" class=\"hidden-answer\" style=\"display: none\">1<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042525428\" class=\"exercise\">\n<div id=\"fs-id1165042525430\" class=\"textbox\">\n<p id=\"fs-id1165042525432\"><strong>39.<\/strong> [latex]\\underset{x\\to 0}{\\lim} (1-\\frac{1}{x})^x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042525481\" class=\"exercise\">\n<div id=\"fs-id1165042525483\" class=\"textbox\">\n<p id=\"fs-id1165042525485\"><strong>40.<\/strong> [latex]\\underset{x\\to \\infty }{\\lim} (1-\\frac{1}{x})^x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042525526\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042525526\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042525526\">[latex]\\frac{1}{e}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042525538\">For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L\u2019H\u00f4pital\u2019s rule to find the limit directly.<\/p>\n<div id=\"fs-id1165042525544\" class=\"exercise\">\n<div class=\"textbox\">\n<p><strong>41. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043382933\" class=\"exercise\">\n<div id=\"fs-id1165043382935\" class=\"textbox\">\n<p id=\"fs-id1165043382937\"><strong>42. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x \\sin (\\frac{1}{x})[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043382979\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043382979\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043382979\">0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043382987\" class=\"exercise\">\n<div id=\"fs-id1165043382989\" class=\"textbox\">\n<p id=\"fs-id1165043382991\"><strong>43. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{x-1}{1- \\cos (\\pi x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043383052\" class=\"exercise\">\n<div id=\"fs-id1165043383054\" class=\"textbox\">\n<p id=\"fs-id1165043383056\"><strong>44. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{e^{x-1}-1}{x-1}[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q855100\">Show Answer<\/span><\/p>\n<div id=\"q855100\" class=\"hidden-answer\" style=\"display: none\">1<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043383121\" class=\"exercise\">\n<div id=\"fs-id1165043383123\" class=\"textbox\">\n<p id=\"fs-id1165043383125\"><strong>45. [T]\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{(x-1)^2}{\\ln x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042710174\" class=\"exercise\">\n<div id=\"fs-id1165042710176\" class=\"textbox\">\n<p id=\"fs-id1165042710178\"><strong>46. [T]\u00a0<\/strong>[latex]\\underset{x\\to \\pi }{\\lim}\\frac{1+ \\cos x}{ \\sin x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042710220\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042710220\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042710220\">0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042710228\" class=\"exercise\">\n<div id=\"fs-id1165042710230\" class=\"textbox\">\n<p id=\"fs-id1165042710232\"><strong>47. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}( \\csc x-\\frac{1}{x})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042710284\" class=\"exercise\">\n<div id=\"fs-id1165042710286\" class=\"textbox\">\n<p id=\"fs-id1165042710288\"><strong>48. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim} \\tan (x^x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042710332\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042710332\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042710332\">[latex]\\tan (1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042710348\" class=\"exercise\">\n<div id=\"fs-id1165042710350\" class=\"textbox\">\n<p id=\"fs-id1165042710352\"><strong>49. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{\\ln x}{ \\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042710404\" class=\"exercise\">\n<div id=\"fs-id1165042710406\" class=\"textbox\">\n<p id=\"fs-id1165042710408\"><strong>50. [T]\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}\\frac{e^x-e^{\u2212x}}{x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042539145\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042539145\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042539145\">2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165042539157\" class=\"definition\">\n<dt>indeterminate forms<\/dt>\n<dd id=\"fs-id1165042539162\">when evaluating a limit, the forms [latex]0\/0[\/latex], [latex]\\infty \/ \\infty[\/latex], [latex]0 \\cdot \\infty[\/latex], [latex]\\infty -\\infty[\/latex], [latex]0^0[\/latex], [latex]\\infty^0[\/latex], and [latex]1^{\\infty}[\/latex] are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042539243\" class=\"definition\">\n<dt>L\u2019H\u00f4pital\u2019s rule<\/dt>\n<dd id=\"fs-id1165042539249\">if [latex]f[\/latex] and [latex]g[\/latex] are differentiable functions over an interval [latex]a[\/latex], except possibly at [latex]a[\/latex], and [latex]\\underset{x\\to a}{\\lim} f(x)=0=\\underset{x\\to a}{\\lim} g(x)[\/latex] or [latex]\\underset{x\\to a}{\\lim} f(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim} g(x)[\/latex] are infinite, then [latex]\\underset{x\\to a}{\\lim}\\frac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex], assuming the limit on the right exists or is [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex]<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2015\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus I. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89\">http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":311,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus I\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/8b89d172-2927-466f-8661-01abc7ccdba4@2.89\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2015","chapter","type-chapter","status-publish","hentry"],"part":1878,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters\/2015","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters\/2015\/revisions"}],"predecessor-version":[{"id":2766,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters\/2015\/revisions\/2766"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/parts\/1878"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapters\/2015\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/media?parent=2015"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2015"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/contributor?post=2015"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/wp-json\/wp\/v2\/license?post=2015"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}