{"id":27931,"date":"2024-08-16T13:00:25","date_gmt":"2024-08-16T13:00:25","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27931"},"modified":"2024-08-18T06:51:36","modified_gmt":"2024-08-18T06:51:36","slug":"the-squeeze-theorem-for-calculating-limits","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/the-squeeze-theorem-for-calculating-limits\/","title":{"rendered":"The Squeeze Theorem for Calculating Limits"},"content":{"rendered":"<p><center><\/p>\n<h1>The Squeeze Theorem for Calculating Limits<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nThis class presents the Squeeze Theorem, a key tool in calculus for evaluating difficult limits using simpler functions that bound from above and below. It offers a graphical explanation and a formal proof, followed by practical examples. The goal is for students to understand how to apply this theorem to calculate limits more efficiently.<\/em><\/p>\n<p><strong>Learning Objectives:<\/strong><br \/>\nUpon completing this class, the student will be able to<\/p>\n<ul style=\"text-align:left;\">\n<li><strong>Understand<\/strong> the utility of the Squeeze Theorem in limit calculation.<\/li>\n<li><strong>Identify<\/strong> functions that can bound a target function to apply the theorem.<\/li>\n<li><strong>Apply<\/strong> the Squeeze Theorem to calculate difficult limits.<\/li>\n<li><strong>Visualize<\/strong> the concept of the Squeeze Theorem graphically.<\/li>\n<li><strong>Prove<\/strong> the Squeeze Theorem formally.<\/li>\n<\/ul>\n<p><strong><u>TABLE OF CONTENTS<\/u>:<\/strong><br \/>\n<a href=\"#1\">Introduction<\/a><br \/>\n<a href=\"#2\">Graphical Idea of the Squeeze Theorem<\/a><br \/>\n<a href=\"#3\">Proof of the Squeeze Theorem<\/a><br \/>\n<a href=\"#4\">Examples<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/24G_qlEwL9M\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Introduction<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=158s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The utility of the Squeeze Theorem lies in its ease of calculating some difficult limits<\/span><\/strong><\/a> through simpler ones. The reason for the name is that, instead of directly calculating the limit of a function as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, another pair of functions is used, one bounding from above and the other from below, whose limit at <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> coincides and is easy to obtain. Since the original function is always between the two, it is like \u00abthe cheese between the two slices of bread.\u00bb<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Graphical Idea of the Squeeze Theorem<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The idea that synthesizes the theorem is actually quite simple.<\/span> <\/strong><\/a>Let&#8217;s suppose we want to calculate a difficult limit<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x \\to x_0}f(x)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">What is usually done is to take all our knowledge of function algebra to try to <strong>simplify it to the point where we can evaluate it<\/strong>. However, sometimes a different approach is much more efficient. Suppose we have a closed interval <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/span> such that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\in I<\/span><\/span> and there also exist two other functions <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m(x)<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M(x)<\/span><\/span> that satisfy the relation<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in I)(m(x)\\leq f(x) \\leq M(x) )<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">And also<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} m(x) = \\lim_{x\\to x_0} M(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Then it will follow that<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">This is what we can see in the following image.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-gjBfVdaLj-k\/YGDXVUQBqDI\/AAAAAAAAEvg\/d2sNJdweVaoB64O5e2qfBxjZGIyIyGOxgCLcBGAsYHQ\/s0\/teorema%2Bdel%2Bsandwich.PNG\" alt=\"Squeeze Theorem\" class=\"alignnone size-full lazyload\" width=\"513\" height=\"407\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-gjBfVdaLj-k\/YGDXVUQBqDI\/AAAAAAAAEvg\/d2sNJdweVaoB64O5e2qfBxjZGIyIyGOxgCLcBGAsYHQ\/s0\/teorema%2Bdel%2Bsandwich.PNG\" alt=\"Squeeze Theorem\" class=\"alignnone size-full lazyload\" width=\"513\" height=\"407\" \/><\/noscript><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Proof of the Squeeze Theorem<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=404s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">To prove the Squeeze Theorem<\/span><\/strong><\/a>, we will follow the following reasoning:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in I<\/span><\/span><\/span>; <strong>Premise<\/strong><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} m(x) = L<\/span><\/span><\/span> ; <strong>Premise<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta_1 \\gt 0) (|x-x_0|\\lt \\delta_1 \\rightarrow |m(x) -L| \\lt \\epsilon )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} M(x) = L<\/span><\/span><\/span> ; <strong>Premise<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta_2 \\gt 0) (|x-x_0|\\lt \\delta_2 \\rightarrow |M(x) -L| \\lt \\epsilon )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x \\in I)(m(x) \\leq f(x) \\leq M(x) )<\/span><\/span><\/span>; <strong>Premise<\/strong><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x \\in I)(m(x) - L \\leq f(x) - L \\leq M(x) - L )<\/span><\/span>; From (4)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(|m(x) -L|\\lt \\epsilon) \\rightarrow (-\\epsilon \\lt m(x) - L \\lt \\epsilon)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(|M(x) -L|\\lt \\epsilon ) \\rightarrow (-\\epsilon \\lt M(x) - L \\lt \\epsilon) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow ( |M(x) -L| \\lt \\epsilon \\wedge |m(x) -L| \\lt \\epsilon ) )<\/span><\/span>; From (2,3)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow ( - \\epsilon \\lt f(x) - L \\lt \\epsilon ) )<\/span><\/span>; From (1,5,6,7,8)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow |f(x) - L| \\lt \\epsilon ) )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = L\\;\\blacksquare<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Examples<\/h2>\n<p style=\"text-align: justify; color: #000000;\">Using the Squeeze Theorem, we can calculate the limit of functions even when we don&#8217;t have their explicit algebraic expression. Here are a couple of examples:<\/p>\n<p style=\"text-align: justify; color: #000000;\">An example of this occurs in the following situation:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{5-2x^2}\\leq f(x) \\leq \\sqrt{5-x^2}<\/span><\/span>, when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1\\leq x\\leq 1<\/span><\/span>. What is the value of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 0}f(x)<\/span><\/span>? <a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=1082s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">Another practical use of the Squeeze Theorem occurs when the limit itself is not evident compared to other simpler limits that bound it from above and below, such as what is obtained when calculating the following case:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>Calculate: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 0}\\dfrac{\\sin(x)}{x}<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=1157s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The Squeeze Theorem for Calculating Limits Summary: This class presents the Squeeze Theorem, a key tool in calculus for evaluating difficult limits using simpler functions that bound from above and below. It offers a graphical explanation and a formal proof, followed by practical examples. The goal is for students to understand how to apply this [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27930,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":32,"footnotes":""},"categories":[854,567],"tags":[],"citadela-post-location":[],"class_list":["post-27931","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-differential-calculus","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Squeeze Theorem for Calculating Limits - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn to calculate limits using the Squeeze Theorem. 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