{"id":1562,"date":"2024-04-18T17:01:31","date_gmt":"2024-04-18T17:01:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1562"},"modified":"2024-08-05T01:36:55","modified_gmt":"2024-08-05T01:36:55","slug":"the-limit-laws-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-limit-laws-fresh-take\/","title":{"raw":"The Limit Laws: Fresh Take","rendered":"The Limit Laws: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Recognize and apply basic limit laws to evaluate the limits of functions, including polynomials and rational functions<\/li>\r\n\t<li>Evaluate the limit of a function by factoring or by using conjugates<\/li>\r\n\t<li>Evaluate the limit of a function by using the squeeze theorem<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Evaluating Limits<\/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\">Limit Laws:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Sum\/Difference: [latex]\\lim_{x \\to a} [f(x) \\pm g(x)] = \\lim_{x \\to a} f(x) \\pm \\lim_{x \\to a} g(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Product: [latex]\\lim_{x \\to a} [f(x) \\cdot g(x)] = \\lim_{x \\to a} f(x) \\cdot \\lim_{x \\to a} g(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Quotient: [latex]\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{\\lim_{x \\to a} f(x)}{\\lim_{x \\to a} g(x)}[\/latex], if [latex]\\lim_{x \\to a} g(x) \\neq 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Power: [latex]\\lim_{x \\to a} [f(x)]^n = [\\lim_{x \\to a} f(x)]^n[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Polynomial and Rational Function Limits:\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]\\lim_{x \\to a} p(x) = p(a)[\/latex] for polynomials<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} \\frac{p(x)}{q(x)} = \\frac{p(a)}{q(a)}[\/latex] for rational functions, if [latex]q(a) \\neq 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Techniques for Indeterminate Forms:\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Factoring and Simplifying:\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Factor polynomials and cancel common terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Rationalizing (Multiplying by Conjugate):\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Used for limits with square roots<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Simplifying Complex Fractions:\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Combine fractions using LCD<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">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]\\frac{0}{0}[\/latex], [latex]\\frac{\\infty}{\\infty}[\/latex], [latex]0 \\cdot \\infty[\/latex], etc.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Require special techniques for evaluation<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571655486\">Use the limit laws to evaluate [latex]\\underset{x\\to 6}{\\lim}(2x-1)\\sqrt{x+4}[\/latex]. In each step, indicate the limit law applied.<\/p>\r\n<p>[reveal-answer q=\"6635113\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"6635113\"]<\/p>\r\n<p id=\"fs-id1170572209920\">Begin by applying the product law.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572094142\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572094142\"]<\/p>\r\n<p id=\"fs-id1170572094142\">[latex]11\\sqrt{10}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571675277\">Evaluate [latex]\\underset{x\\to -2}{\\lim}(3x^3-2x+7)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"4482011\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4482011\"]<\/p>\r\n<p id=\"fs-id1170571688063\">Use\u00a0limits of polynomial and rational functions<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571688072\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571688072\"]<\/p>\r\n<p id=\"fs-id1170571688072\">[latex]\u221213[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571598007\">Evaluate [latex]\\underset{x\\to -3}{\\lim}\\dfrac{x^2+4x+3}{x^2-9}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571598067\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571598067\"]<\/p>\r\n<p id=\"fs-id1170571598067\">[latex]\\dfrac{1}{3}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571611956\">Evaluate [latex]\\underset{x\\to 5}{\\lim}\\dfrac{\\sqrt{x-1}-2}{x-5}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571612008\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571612008\"]<\/p>\r\n<p id=\"fs-id1170571612008\">[latex]\\dfrac{1}{4}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572394360\">Evaluate [latex]\\underset{x\\to -3}{\\lim}\\dfrac{\\frac{1}{x+2}+1}{x+3}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571648126\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571648126\"]<\/p>\r\n<p id=\"fs-id1170571648126\">[latex]\u22121[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571681429\">Evaluate [latex]\\underset{x\\to 3}{\\lim}\\left(\\dfrac{1}{x-3}-\\dfrac{4}{x^2-2x-3}\\right)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"80944622\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"80944622\"]<\/p>\r\n<p id=\"fs-id1170572233797\">Don\u2019t forget to factor [latex]x^2-2x-3[\/latex] before getting a common denominator.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572233826\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572233826\"]<\/p>\r\n<p id=\"fs-id1170572233826\">[latex]\\frac{1}{4}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>The Squeeze Theorem<\/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\">The Squeeze Theorem:\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]g(x) \\leq f(x) \\leq h(x)[\/latex] near [latex]a[\/latex], and [latex]\\lim_{x \\to a} g(x) = \\lim_{x \\to a} h(x) = L[\/latex], then [latex]\\lim_{x \\to a} f(x) = L[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Trigonometric Limits:\u00a0\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\lim_{\\theta \\to 0} \\sin \\theta = 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\lim_{\\theta \\to 0} \\cos \\theta = 1[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\lim_{\\theta \\to 0} \\frac{\\sin \\theta}{\\theta} = 1[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\lim_{\\theta \\to 0} \\frac{1 - \\cos \\theta}{\\theta} = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Geometric Intuition:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For small [latex]\\theta[\/latex]: [latex]0 &lt; \\sin \\theta &lt; \\theta &lt; \\tan \\theta[\/latex] (in radians)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">It is helpful to visualize the unit circle for sine, cosine, and tangent<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Evaluate the following limit using the Squeeze Theorem:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\lim_{\\theta \\to 0} \\frac{\\theta - \\sin \\theta}{\\theta^3}[\/latex]<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"384919\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"384919\"]<\/p>\r\n<p>Recall that for small [latex]\\theta[\/latex]: [latex]0 &lt; \\sin \\theta &lt; \\theta[\/latex] This means: [latex]0 &lt; \\theta - \\sin \\theta &lt; \\theta - 0 = \\theta[\/latex]<\/p>\r\n<p>Divide all parts by [latex]\\theta^3[\/latex] (which is positive for small, positive [latex]\\theta[\/latex]):<\/p>\r\n<p style=\"text-align: center;\">[latex]0 &lt; \\frac{\\theta - \\sin \\theta}{\\theta^3} &lt; \\frac{\\theta}{\\theta^3} = \\frac{1}{\\theta^2}[\/latex]<\/p>\r\n<p>Now we have bounds for our function:<\/p>\r\n<p style=\"text-align: center;\">[latex]0 \\leq \\frac{\\theta - \\sin \\theta}{\\theta^3} \\leq \\frac{1}{\\theta^2}[\/latex]<\/p>\r\n<p>As [latex]\\theta \\to 0[\/latex], both bounds approach [latex]0[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{\\theta \\to 0} 0 = 0[\/latex] and [latex]\\lim_{\\theta \\to 0} \\frac{1}{\\theta^2} = \\infty[\/latex]<\/p>\r\n<p>By the Squeeze Theorem:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{\\theta \\to 0} \\frac{\\theta - \\sin \\theta}{\\theta^3} = 0[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Evaluate the following limit:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\lim_{x \\to 0} x \\cos(\\frac{1}{x})[\/latex]<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"446513\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"446513\"]<\/p>\r\n<p>Note that [latex]\\cos(\\frac{1}{x})[\/latex] oscillates between [latex]-1[\/latex] and [latex]1[\/latex] as [latex]x[\/latex] approaches [latex]0[\/latex].<\/p>\r\n<p>Therefore, we can bound our function:<\/p>\r\n<p style=\"text-align: center;\">[latex]-|x| \\leq x \\cos(\\frac{1}{x}) \\leq |x|[\/latex]<\/p>\r\n<p>As [latex]x[\/latex] approaches [latex]0[\/latex], both [latex]-|x|[\/latex] and [latex]|x|[\/latex] approach [latex]0[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{x \\to 0} -|x| = \\lim_{x \\to 0} |x| = 0[\/latex]<\/p>\r\n<p>By the Squeeze Theorem:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{x \\to 0} x \\cos(\\frac{1}{x}) = 0[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize and apply basic limit laws to evaluate the limits of functions, including polynomials and rational functions<\/li>\n<li>Evaluate the limit of a function by factoring or by using conjugates<\/li>\n<li>Evaluate the limit of a function by using the squeeze theorem<\/li>\n<\/ul>\n<\/section>\n<h2>Evaluating Limits<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Limit Laws:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sum\/Difference: [latex]\\lim_{x \\to a} [f(x) \\pm g(x)] = \\lim_{x \\to a} f(x) \\pm \\lim_{x \\to a} g(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Product: [latex]\\lim_{x \\to a} [f(x) \\cdot g(x)] = \\lim_{x \\to a} f(x) \\cdot \\lim_{x \\to a} g(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Quotient: [latex]\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{\\lim_{x \\to a} f(x)}{\\lim_{x \\to a} g(x)}[\/latex], if [latex]\\lim_{x \\to a} g(x) \\neq 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Power: [latex]\\lim_{x \\to a} [f(x)]^n = [\\lim_{x \\to a} f(x)]^n[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Polynomial and Rational Function Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} p(x) = p(a)[\/latex] for polynomials<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} \\frac{p(x)}{q(x)} = \\frac{p(a)}{q(a)}[\/latex] for rational functions, if [latex]q(a) \\neq 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Techniques for Indeterminate Forms:\n<ul>\n<li class=\"whitespace-normal break-words\">Factoring and Simplifying:\n<ul>\n<li class=\"whitespace-normal break-words\">Factor polynomials and cancel common terms<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Rationalizing (Multiplying by Conjugate):\n<ul>\n<li class=\"whitespace-normal break-words\">Used for limits with square roots<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplifying Complex Fractions:\n<ul>\n<li class=\"whitespace-normal break-words\">Combine fractions using LCD<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Indeterminate Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{0}{0}[\/latex], [latex]\\frac{\\infty}{\\infty}[\/latex], [latex]0 \\cdot \\infty[\/latex], etc.<\/li>\n<li class=\"whitespace-normal break-words\">Require special techniques for evaluation<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571655486\">Use the limit laws to evaluate [latex]\\underset{x\\to 6}{\\lim}(2x-1)\\sqrt{x+4}[\/latex]. In each step, indicate the limit law applied.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q6635113\">Hint<\/button><\/p>\n<div id=\"q6635113\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572209920\">Begin by applying the product law.<\/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-id1170572094142\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572094142\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572094142\">[latex]11\\sqrt{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571675277\">Evaluate [latex]\\underset{x\\to -2}{\\lim}(3x^3-2x+7)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4482011\">Hint<\/button><\/p>\n<div id=\"q4482011\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571688063\">Use\u00a0limits of polynomial and rational functions<\/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-id1170571688072\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571688072\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571688072\">[latex]\u221213[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571598007\">Evaluate [latex]\\underset{x\\to -3}{\\lim}\\dfrac{x^2+4x+3}{x^2-9}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571598067\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571598067\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571598067\">[latex]\\dfrac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571611956\">Evaluate [latex]\\underset{x\\to 5}{\\lim}\\dfrac{\\sqrt{x-1}-2}{x-5}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571612008\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571612008\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571612008\">[latex]\\dfrac{1}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572394360\">Evaluate [latex]\\underset{x\\to -3}{\\lim}\\dfrac{\\frac{1}{x+2}+1}{x+3}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571648126\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571648126\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571648126\">[latex]\u22121[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571681429\">Evaluate [latex]\\underset{x\\to 3}{\\lim}\\left(\\dfrac{1}{x-3}-\\dfrac{4}{x^2-2x-3}\\right)[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q80944622\">Hint<\/button><\/p>\n<div id=\"q80944622\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572233797\">Don\u2019t forget to factor [latex]x^2-2x-3[\/latex] before getting a common denominator.<\/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-id1170572233826\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572233826\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572233826\">[latex]\\frac{1}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>The Squeeze Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">The Squeeze Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]g(x) \\leq f(x) \\leq h(x)[\/latex] near [latex]a[\/latex], and [latex]\\lim_{x \\to a} g(x) = \\lim_{x \\to a} h(x) = L[\/latex], then [latex]\\lim_{x \\to a} f(x) = L[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Trigonometric Limits:\u00a0\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{\\theta \\to 0} \\sin \\theta = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{\\theta \\to 0} \\cos \\theta = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{\\theta \\to 0} \\frac{\\sin \\theta}{\\theta} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{\\theta \\to 0} \\frac{1 - \\cos \\theta}{\\theta} = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Intuition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For small [latex]\\theta[\/latex]: [latex]0 < \\sin \\theta < \\theta < \\tan \\theta[\/latex] (in radians)<\/li>\n<li class=\"whitespace-normal break-words\">It is helpful to visualize the unit circle for sine, cosine, and tangent<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Evaluate the following limit using the Squeeze Theorem:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\lim_{\\theta \\to 0} \\frac{\\theta - \\sin \\theta}{\\theta^3}[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q384919\">Show Answer<\/button><\/p>\n<div id=\"q384919\" class=\"hidden-answer\" style=\"display: none\">\n<p>Recall that for small [latex]\\theta[\/latex]: [latex]0 < \\sin \\theta < \\theta[\/latex] This means: [latex]0 < \\theta - \\sin \\theta < \\theta - 0 = \\theta[\/latex]<\/p>\n<p>Divide all parts by [latex]\\theta^3[\/latex] (which is positive for small, positive [latex]\\theta[\/latex]):<\/p>\n<p style=\"text-align: center;\">[latex]0 < \\frac{\\theta - \\sin \\theta}{\\theta^3} < \\frac{\\theta}{\\theta^3} = \\frac{1}{\\theta^2}[\/latex]<\/p>\n<p>Now we have bounds for our function:<\/p>\n<p style=\"text-align: center;\">[latex]0 \\leq \\frac{\\theta - \\sin \\theta}{\\theta^3} \\leq \\frac{1}{\\theta^2}[\/latex]<\/p>\n<p>As [latex]\\theta \\to 0[\/latex], both bounds approach [latex]0[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{\\theta \\to 0} 0 = 0[\/latex] and [latex]\\lim_{\\theta \\to 0} \\frac{1}{\\theta^2} = \\infty[\/latex]<\/p>\n<p>By the Squeeze Theorem:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{\\theta \\to 0} \\frac{\\theta - \\sin \\theta}{\\theta^3} = 0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Evaluate the following limit:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\lim_{x \\to 0} x \\cos(\\frac{1}{x})[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q446513\">Show Answer<\/button><\/p>\n<div id=\"q446513\" class=\"hidden-answer\" style=\"display: none\">\n<p>Note that [latex]\\cos(\\frac{1}{x})[\/latex] oscillates between [latex]-1[\/latex] and [latex]1[\/latex] as [latex]x[\/latex] approaches [latex]0[\/latex].<\/p>\n<p>Therefore, we can bound our function:<\/p>\n<p style=\"text-align: center;\">[latex]-|x| \\leq x \\cos(\\frac{1}{x}) \\leq |x|[\/latex]<\/p>\n<p>As [latex]x[\/latex] approaches [latex]0[\/latex], both [latex]-|x|[\/latex] and [latex]|x|[\/latex] approach [latex]0[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{x \\to 0} -|x| = \\lim_{x \\to 0} |x| = 0[\/latex]<\/p>\n<p>By the Squeeze Theorem:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{x \\to 0} x \\cos(\\frac{1}{x}) = 0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1562"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1562\/revisions"}],"predecessor-version":[{"id":3680,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1562\/revisions\/3680"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1562\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1562"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1562"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1562"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1562"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}