{"id":3140,"date":"2024-06-13T16:32:51","date_gmt":"2024-06-13T16:32:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3140"},"modified":"2024-08-05T02:29:37","modified_gmt":"2024-08-05T02:29:37","slug":"lhopitals-rule-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/lhopitals-rule-fresh-take\/","title":{"raw":"L\u2019H\u00f4pital\u2019s Rule: Fresh Take","rendered":"L\u2019H\u00f4pital\u2019s Rule: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Spot indeterminate forms like in calculations, and use L\u2019H\u00f4pital\u2019s rule to find precise values<\/li>\r\n\t<li>Explain how quickly different functions increase or decrease compared to each other<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div id=\"fs-id1165043085155\" class=\"bc-section section\">\r\n<h2>L\u2019H\u00f4pital\u2019s Rule<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Purpose:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Evaluates limits that are otherwise challenging to compute directly<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Resolves indeterminate forms of type [latex]\\frac{0}{0}[\/latex] and [latex]\\frac{\\infty}{\\infty}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The Rule (0\/0 case):\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If [latex]\\lim_{x \\to a} f(x) = 0[\/latex] and [latex]\\lim_{x \\to a} g(x) = 0[\/latex], then: [latex]\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The Rule (\u221e\/\u221e case):\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If [latex]\\lim_{x \\to a} f(x) = \\pm\\infty[\/latex] and [latex]\\lim_{x \\to a} g(x) = \\pm\\infty[\/latex], then: [latex]\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applicability:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Works for one-sided limits<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applies when [latex]a = \\infty[\/latex] or [latex]-\\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Can be applied repeatedly if needed<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Limitations:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Only applicable to indeterminate forms [latex]\\frac{0}{0}[\/latex] and [latex]\\frac{\\infty}{\\infty}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Not applicable when limits of numerator and denominator are finite and non-zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Evaluate [latex]\\underset{x\\to 0}{\\lim}\\dfrac{x}{\\tan x}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"37002811\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"37002811\"]<\/p>\r\n<p>[latex]\\frac{d}{dx} \\tan x= \\sec ^2 x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042377480\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042377480\"]<\/p>\r\n<p>[latex]1[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{\\ln x}{5x}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"30011179\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"30011179\"]<\/p>\r\n<p>[latex]\\frac{d}{dx}\\ln x=\\frac{1}{x}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042367881\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042367881\"]<\/p>\r\n<p>[latex]0[\/latex]<br \/>\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Other Indeterminate Forms<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Additional Indeterminate Forms:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]0 \\cdot \\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\infty - \\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]1^{\\infty}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\infty^0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]0^0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Strategy for L'H\u00f4pital's Rule:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Rewrite the expression to create a fraction of the form [latex]\\frac{0}{0}[\/latex] or [latex]\\frac{\\infty}{\\infty}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply L'H\u00f4pital's Rule to the resulting fraction<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Handling [latex]0 \\cdot \\infty[\/latex] Form:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Rewrite as a fraction, typically [latex]\\frac{\\text{finite term}}{\\frac{1}{\\text{infinite term}}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Handling [latex]\\infty - \\infty[\/latex] Form:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Combine terms over a common denominator<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Handling Exponential Forms:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Use logarithms to convert to a product or quotient<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply L'H\u00f4pital's Rule to the resulting expression<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Evaluate [latex]\\underset{x\\to \\infty}{\\lim} x^{\\frac{1}{\\ln x}}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"3762844\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3762844\"]<\/p>\r\n<p>Let [latex]y=x^{1\/ \\ln x}[\/latex] and apply the natural logarithm to both sides of the equation.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043108248\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043108248\"]<\/p>\r\n<p>[latex]e[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Evaluate [latex]\\underset{x\\to 0^+}{\\lim} x^x[\/latex]<\/p>\r\n<p>[reveal-answer q=\"929037\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"929037\"]<\/p>\r\n<p>Let [latex]y=x^x[\/latex] and take the natural logarithm of both sides of the equation.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042660293\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042660293\"]<\/p>\r\n<p>[latex]1[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Growth Rates of Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Comparing Growth Rates:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Functions can approach infinity at different rates<\/li>\r\n\t<li class=\"whitespace-normal break-words\">We compare their relative growth as [latex]x \\to \\infty[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Defining Faster Growth:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]g(x)[\/latex] grows faster than [latex]f(x)[\/latex] if: [latex]\\lim_{x \\to \\infty} \\frac{g(x)}{f(x)} = \\infty[\/latex] or [latex]\\lim_{x \\to \\infty} \\frac{f(x)}{g(x)} = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Same Growth Rate:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] and [latex]g(x)[\/latex] grow at the same rate if: [latex]\\lim_{x \\to \\infty} \\frac{f(x)}{g(x)} = M[\/latex], where [latex]M[\/latex] is a non-zero constant<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Hierarchy of Growth Rates:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Exponential &gt; Power &gt; Logarithmic<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]e^x[\/latex] grows faster than [latex]x^p[\/latex] for any [latex]p &gt; 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]x^p[\/latex] grows faster than [latex]\\ln x[\/latex] for any [latex]p &gt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Compare the growth rates of [latex]x^{100}[\/latex] and [latex]2^x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"41567703\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"41567703\"]<\/p>\r\n<p>Apply L\u2019H\u00f4pital\u2019s rule to [latex]\\frac{x^{100}}{2^x}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042463749\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042463749\"]<\/p>\r\n<p>The function [latex]2^x[\/latex] grows faster than [latex]x^{100}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Spot indeterminate forms like in calculations, and use L\u2019H\u00f4pital\u2019s rule to find precise values<\/li>\n<li>Explain how quickly different functions increase or decrease compared to each other<\/li>\n<\/ul>\n<\/section>\n<div id=\"fs-id1165043085155\" class=\"bc-section section\">\n<h2>L\u2019H\u00f4pital\u2019s Rule<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Purpose:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Evaluates limits that are otherwise challenging to compute directly<\/li>\n<li class=\"whitespace-normal break-words\">Resolves indeterminate forms of type [latex]\\frac{0}{0}[\/latex] and [latex]\\frac{\\infty}{\\infty}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The Rule (0\/0 case):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]\\lim_{x \\to a} f(x) = 0[\/latex] and [latex]\\lim_{x \\to a} g(x) = 0[\/latex], then: [latex]\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The Rule (\u221e\/\u221e case):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]\\lim_{x \\to a} f(x) = \\pm\\infty[\/latex] and [latex]\\lim_{x \\to a} g(x) = \\pm\\infty[\/latex], then: [latex]\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applicability:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Works for one-sided limits<\/li>\n<li class=\"whitespace-normal break-words\">Applies when [latex]a = \\infty[\/latex] or [latex]-\\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Can be applied repeatedly if needed<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Limitations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Only applicable to indeterminate forms [latex]\\frac{0}{0}[\/latex] and [latex]\\frac{\\infty}{\\infty}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Not applicable when limits of numerator and denominator are finite and non-zero<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Evaluate [latex]\\underset{x\\to 0}{\\lim}\\dfrac{x}{\\tan x}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q37002811\">Hint<\/button><\/p>\n<div id=\"q37002811\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{d}{dx} \\tan x= \\sec ^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042377480\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042377480\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]1[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate [latex]\\underset{x\\to \\infty }{\\lim}\\dfrac{\\ln x}{5x}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q30011179\">Hint<\/button><\/p>\n<div id=\"q30011179\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{d}{dx}\\ln x=\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042367881\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042367881\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]0[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<h2>Other Indeterminate Forms<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Additional Indeterminate Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]0 \\cdot \\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\infty - \\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1^{\\infty}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\infty^0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0^0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Strategy for L&#8217;H\u00f4pital&#8217;s Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rewrite the expression to create a fraction of the form [latex]\\frac{0}{0}[\/latex] or [latex]\\frac{\\infty}{\\infty}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply L&#8217;H\u00f4pital&#8217;s Rule to the resulting fraction<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Handling [latex]0 \\cdot \\infty[\/latex] Form:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rewrite as a fraction, typically [latex]\\frac{\\text{finite term}}{\\frac{1}{\\text{infinite term}}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Handling [latex]\\infty - \\infty[\/latex] Form:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combine terms over a common denominator<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Handling Exponential Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use logarithms to convert to a product or quotient<\/li>\n<li class=\"whitespace-normal break-words\">Apply L&#8217;H\u00f4pital&#8217;s Rule to the resulting expression<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Evaluate [latex]\\underset{x\\to \\infty}{\\lim} x^{\\frac{1}{\\ln x}}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3762844\">Hint<\/button><\/p>\n<div id=\"q3762844\" class=\"hidden-answer\" style=\"display: none\">\n<p>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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043108248\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043108248\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]e[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate [latex]\\underset{x\\to 0^+}{\\lim} x^x[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q929037\">Hint<\/button><\/p>\n<div id=\"q929037\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]y=x^x[\/latex] and take the natural logarithm of both sides of the equation.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042660293\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042660293\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Growth Rates of Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Comparing Growth Rates:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Functions can approach infinity at different rates<\/li>\n<li class=\"whitespace-normal break-words\">We compare their relative growth as [latex]x \\to \\infty[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Defining Faster Growth:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]g(x)[\/latex] grows faster than [latex]f(x)[\/latex] if: [latex]\\lim_{x \\to \\infty} \\frac{g(x)}{f(x)} = \\infty[\/latex] or [latex]\\lim_{x \\to \\infty} \\frac{f(x)}{g(x)} = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Same Growth Rate:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] and [latex]g(x)[\/latex] grow at the same rate if: [latex]\\lim_{x \\to \\infty} \\frac{f(x)}{g(x)} = M[\/latex], where [latex]M[\/latex] is a non-zero constant<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Hierarchy of Growth Rates:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exponential &gt; Power &gt; Logarithmic<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e^x[\/latex] grows faster than [latex]x^p[\/latex] for any [latex]p > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x^p[\/latex] grows faster than [latex]\\ln x[\/latex] for any [latex]p > 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Compare the growth rates of [latex]x^{100}[\/latex] and [latex]2^x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q41567703\">Hint<\/button><\/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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042463749\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042463749\" class=\"hidden-answer\" style=\"display: none\">\n<p>The function [latex]2^x[\/latex] grows faster than 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