{"id":415,"date":"2021-02-04T02:05:09","date_gmt":"2021-02-04T02:05:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=415"},"modified":"2022-03-16T05:50:45","modified_gmt":"2022-03-16T05:50:45","slug":"growth-rates-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/growth-rates-of-functions\/","title":{"raw":"Growth Rates of Functions","rendered":"Growth Rates of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Describe the relative growth rates of functions<\/li>\r\n<\/ul>\r\n<\/div>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{x^3}{x^2}=\\underset{x\\to \\infty}{\\lim} x=\\infty[\/latex]\u00a0 or, equivalently, [latex]\\underset{x\\to \\infty}{\\lim}\\dfrac{x^2}{x^3}=\\underset{x\\to \\infty }{\\lim}\\dfrac{1}{x}=0[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{x^2}{3x^2+4x+1}=\\dfrac{1}{3}[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{g(x)}{f(x)}=\\infty[\/latex]\u00a0 or, equivalently, [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{f(x)}{g(x)}=0[\/latex]<\/div>\r\n&nbsp;\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{f(x)}{g(x)}=M[\/latex],<\/div>\r\n&nbsp;\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=\"textbook exercises\">\r\n<h3>example: Comparing the Growth Rates of [latex]\\ln x[\/latex], [latex]x^2[\/latex], and [latex]e^x[\/latex]<\/h3>\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}\\left(\\dfrac{f(x)}{g(x)}\\right)[\/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 id=\"fs-id1165043422464\" class=\"exercise\">\r\n\r\n[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= \\infty[\/latex], we can use L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\left[\\frac{x^2}{e^x}\\right][\/latex]. We obtain\r\n<div id=\"fs-id1165042631803\" class=\"equation unnumbered\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[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 Figure 3 and the table below).\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\" \/> Figure 3. An exponential function grows at a faster rate than a power function.[\/caption]\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\" style=\"text-align: center;\">[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 Figure 4 and the table below).\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\" \/> Figure 4. A power function grows at a faster rate than a logarithmic function.[\/caption]\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\nWatch the following video to see the worked solution to Example: Comparing the Growth Rates of [latex]\\ln x[\/latex], [latex]x^2[\/latex], and [latex]e^x[\/latex].\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/e58sGIZe1wU?controls=0&amp;start=1081&amp;end=1246&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.8LHopitalsRule1081to1246_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.8 L'Hopital's Rule\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1165042463715\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nCompare the growth rates of [latex]x^{100}[\/latex] and [latex]2^x[\/latex].\r\n\r\n[reveal-answer q=\"41567703\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"41567703\"]\r\n\r\nApply L\u2019H\u00f4pital\u2019s rule to [latex]\\frac{x^{100}}{2^x}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165042463749\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042463749\"]\r\n\r\nThe function [latex]2^x[\/latex] grows faster than [latex]x^{100}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165042463803\">Using the same ideas as in the last example. 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 Figure 5 and the table below it, 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[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\" \/> Figure 5. 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]\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 Figure 6 and the table below it, we compare [latex]\\ln x[\/latex] with [latex]\\sqrt[3]{x}[\/latex] and [latex]\\sqrt{x}[\/latex].<\/p>\r\n\r\n[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\" \/> Figure 6. 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]\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><span style=\"font-size: 16px; font-weight: 400;\">A logarithmic function grows at a slower rate than any root function<\/span><\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><span style=\"font-size: 16px;\">[latex]x[\/latex]<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">10<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">100<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">1000<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">10,000<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><span style=\"font-size: 16px;\">[latex]\\ln x[\/latex]<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">2.303<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">4.605<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">6.908<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">9.210<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><span style=\"font-size: 16px;\">[latex]\\sqrt[3]{x}[\/latex]<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">2.154<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">4.642<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">10<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">21.544<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><span style=\"font-size: 16px;\">[latex]\\sqrt{x}[\/latex]<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">3.162<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">10<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">31.623<\/span><\/td>\r\n<td><span style=\"font-size: 16px;\">100<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Describe the relative growth rates of functions<\/li>\n<\/ul>\n<\/div>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{x^3}{x^2}=\\underset{x\\to \\infty}{\\lim} x=\\infty[\/latex]\u00a0 or, equivalently, [latex]\\underset{x\\to \\infty}{\\lim}\\dfrac{x^2}{x^3}=\\underset{x\\to \\infty }{\\lim}\\dfrac{1}{x}=0[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{x^2}{3x^2+4x+1}=\\dfrac{1}{3}[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{g(x)}{f(x)}=\\infty[\/latex]\u00a0 or, equivalently, [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{f(x)}{g(x)}=0[\/latex]<\/div>\n<p>&nbsp;<\/p>\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\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{f(x)}{g(x)}=M[\/latex],<\/div>\n<p>&nbsp;<\/p>\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=\"textbook exercises\">\n<h3>example: Comparing the Growth Rates of [latex]\\ln x[\/latex], [latex]x^2[\/latex], and [latex]e^x[\/latex]<\/h3>\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}\\left(\\dfrac{f(x)}{g(x)}\\right)[\/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 id=\"fs-id1165043422464\" class=\"exercise\">\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= \\infty[\/latex], we can use L\u2019H\u00f4pital\u2019s rule to evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\left[\\frac{x^2}{e^x}\\right][\/latex]. We obtain\n<div id=\"fs-id1165042631803\" class=\"equation unnumbered\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[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\" style=\"text-align: center;\">[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 Figure 3 and the table below).<\/p>\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\">Figure 3. An exponential function grows at a faster rate than a power function.<\/p>\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\" style=\"text-align: center;\">[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 Figure 4 and the table below).<\/p>\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\">Figure 4. A power function grows at a faster rate than a logarithmic function.<\/p>\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<p>Watch the following video to see the worked solution to Example: Comparing the Growth Rates of [latex]\\ln x[\/latex], [latex]x^2[\/latex], and [latex]e^x[\/latex].<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/e58sGIZe1wU?controls=0&amp;start=1081&amp;end=1246&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.8LHopitalsRule1081to1246_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.8 L&#8217;Hopital&#8217;s Rule&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1165042463715\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Compare the growth rates of [latex]x^{100}[\/latex] and [latex]2^x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q41567703\">Hint<\/span><\/p>\n<div id=\"q41567703\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply L\u2019H\u00f4pital\u2019s rule to [latex]\\frac{x^{100}}{2^x}[\/latex]<\/p>\n<\/div>\n<\/div>\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>The function [latex]2^x[\/latex] grows faster than [latex]x^{100}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042463803\">Using the same ideas as in the last example. 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 Figure 5 and the table below it, 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 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\">Figure 5. 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<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 Figure 6 and the table below it, we compare [latex]\\ln x[\/latex] with [latex]\\sqrt[3]{x}[\/latex] and [latex]\\sqrt{x}[\/latex].<\/p>\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\">Figure 6. 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<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><span style=\"font-size: 16px; font-weight: 400;\">A logarithmic function grows at a slower rate than any root function<\/span><\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><span style=\"font-size: 16px;\">[latex]x[\/latex]<\/span><\/td>\n<td><span style=\"font-size: 16px;\">10<\/span><\/td>\n<td><span style=\"font-size: 16px;\">100<\/span><\/td>\n<td><span style=\"font-size: 16px;\">1000<\/span><\/td>\n<td><span style=\"font-size: 16px;\">10,000<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><span style=\"font-size: 16px;\">[latex]\\ln x[\/latex]<\/span><\/td>\n<td><span style=\"font-size: 16px;\">2.303<\/span><\/td>\n<td><span style=\"font-size: 16px;\">4.605<\/span><\/td>\n<td><span style=\"font-size: 16px;\">6.908<\/span><\/td>\n<td><span style=\"font-size: 16px;\">9.210<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><span style=\"font-size: 16px;\">[latex]\\sqrt[3]{x}[\/latex]<\/span><\/td>\n<td><span style=\"font-size: 16px;\">2.154<\/span><\/td>\n<td><span style=\"font-size: 16px;\">4.642<\/span><\/td>\n<td><span style=\"font-size: 16px;\">10<\/span><\/td>\n<td><span style=\"font-size: 16px;\">21.544<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><span style=\"font-size: 16px;\">[latex]\\sqrt{x}[\/latex]<\/span><\/td>\n<td><span style=\"font-size: 16px;\">3.162<\/span><\/td>\n<td><span style=\"font-size: 16px;\">10<\/span><\/td>\n<td><span style=\"font-size: 16px;\">31.623<\/span><\/td>\n<td><span style=\"font-size: 16px;\">100<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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-415\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>4.8 L&#039;Hopital&#039;s Rule. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/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>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/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":17533,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"4.8 L\\'Hopital\\'s Rule\",\"author\":\"Ryan 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