{"id":72544,"date":"2021-05-07T08:28:35","date_gmt":"2021-05-07T13:28:35","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=72544"},"modified":"2026-07-20T15:13:30","modified_gmt":"2026-07-20T20:13:30","slug":"squeeze-theorem","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/squeeze-theorem\/","title":{"rendered":"Squeeze Theorem"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_CJlrGw250fE\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_CJlrGw250fE\" data-source-videoID=\"CJlrGw250fE\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Squeeze Theorem Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Squeeze Theorem\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_CJlrGw250fE:hover {cursor:pointer;} img#videoThumbnailImage_CJlrGw250fE {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1752-thumb-final-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_CJlrGw250fE\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_CJlrGw250fE\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Squeeze Theorem\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_CJlrGw250fE\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_CJlrGw250fE\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_CJlrGw250fE\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction eG1I_Function() {\n  var x = document.getElementById(\"eG1I\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"eG1I_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"eG1I\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_the_Squeeze_Theorem\" class=\"smooth-scroll\">What is the Squeeze Theorem?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_1\" class=\"smooth-scroll\">Example #1<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_2\" class=\"smooth-scroll\">Example #2<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_3\" class=\"smooth-scroll\">Example #3<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_4\" class=\"smooth-scroll\">Example #4<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Squeeze_Theorem_Practice_Questions\" class=\"smooth-scroll\">Squeeze Theorem Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello and welcome to this video about the squeeze theorem!<\/p>\n<h2><span id=\"What_is_the_Squeeze_Theorem\" class=\"m-toc-anchor\"><\/span>What is the Squeeze Theorem?<\/h2>\n<p>\nWhen calculating limits, occasionally we run into examples that cannot be evaluated by the more \u201cconventional\u201d methods\u2014direct substitution is inconclusive, algebra proves fruitless, even graphing may not give a clear picture.<\/p>\n<p>Sometimes, we can actually squeeze a function between two other functions to determine a limit.<\/p>\n<h3><span id=\"Other_Names\" class=\"m-toc-anchor\"><\/span>Other Names<\/h3>\n<p>\nFor this reason, the theorem is also referred to as:<\/p>\n<ul>\n<li>The sandwich theorem<\/li>\n<li>The sandwich rule<\/li>\n<li>The police theorem<\/li>\n<li>The pinching theorem<\/li>\n<li>The squeeze lemma<\/li>\n<li>The theorem of carabinieri (in Italy)<\/li>\n<\/ul>\n<h2><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h2>\n<p>\nLet\u2019s see it in action with a familiar limit we know before stating it formally. We know: <\/p>\n<div class=\"examplesentence\" style=\"font-size: 105%;\">\\[\\lim\\limits_{x \\to 0} \\frac{\\sin x}{x} = 1\\]<\/div>\n<p>\n&nbsp;<br \/>\nSubstituting 0 yields the indeterminate form \\(\\frac{0}{0}\\), but we can\u2019t use algebra to find the limit because the numerator and denominator can\u2019t be simplified. We can see the limit on a graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-72553\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-2.png\" alt=\"\" width=\"1334\" height=\"692\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-2.png 1334w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-2-300x156.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-2-1024x531.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-2-768x398.png 768w\" sizes=\"auto, (max-width: 1334px) 100vw, 1334px\" \/><\/p>\n<p>We can use L\u2019H\u00f4pital\u2019s Rule:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 105%;\">\n<math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>sin<\/mi><mo>&#x2061;<\/mo><mi>x<\/mi><\/mrow><mi>x<\/mi><\/mfrac><mo>=<\/mo><mspace width=\"0.3em\"\/><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mfrac><mo>d<\/mo><mi>dx<\/mi><\/mfrac><mo>(<\/mo><mi>sin<\/mi><mo>&#x2061;<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><mrow><mfrac><mo>d<\/mo><mi>dx<\/mi><\/mfrac><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><\/mfrac><\/mrow><\/math><math display=\"inline\"><mrow><mo>=<\/mo><mspace width=\"0.3em\"\/><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mi>cos<\/mi><mn>1<\/mn><\/mfrac><mo>=<\/mo><mi>cos<\/mi><mo>&#x2061;<\/mo><mn>0<\/mn><mo>=<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/div>\n<p>\n&nbsp;<br \/>\nAnother option is to use the squeeze theorem. This graph shows that \\(\\frac{\\text{sin }x}{x}\\) is between the cosine function and the horizontal line \\(y=1\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-72556\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-4.png\" alt=\"\" width=\"798\" height=\"326\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-4.png 798w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-4-300x123.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-4-768x314.png 768w\" sizes=\"auto, (max-width: 798px) 100vw, 798px\" \/><\/p>\n<p>When looking for the \u201cbetween-ness,\u201d we\u2019ll zoom in close to \\(x = 0\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-72559\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-5.png\" alt=\"\" width=\"711\" height=\"404\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-5.png 711w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-5-300x170.png 300w\" sizes=\"auto, (max-width: 711px) 100vw, 711px\" \/><\/p>\n<p>No matter how close we zoom in to the graph, \\(\\frac{\\text{sin }x}{x}\\) will always be between \\(\\text{cos }x\\) and 1. By between, we are referring to the function outputs. Let\u2019s zoom in numerically. Of course we can\u2019t use 0, but we can use numbers really close to 0:<\/p>\n<div class=\"ATable-container\">\n<table class=\"ATable\" style=\"margin: auto; width: 70%\">\n<tbody>\n<tr>\n<td>\\(x\\)<\/td>\n<td>\u22120.1<\/td>\n<td>\u22120.01<\/td>\n<td>0.01<\/td>\n<td>0.1<\/td>\n<\/tr>\n<tr>\n<td>\\(h(x)=1\\)<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>\\(g(x)=\\frac{\\text{sin }x}{x}\\)<\/td>\n<td>0.998<\/td>\n<td>0.99998<\/td>\n<td>0.99998<\/td>\n<td>0.998<\/td>\n<\/tr>\n<tr>\n<td>\\(f(x)=\\text{cos }x\\)<\/td>\n<td>0.995<\/td>\n<td>0.99995<\/td>\n<td>0.99995<\/td>\n<td>0.995<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\n&nbsp;<br \/>\nNo matter how close to 0 we get, the values of \\(g(x)\\) are always between the values of \\(f(x)\\) and \\(h(x)\\).<\/p>\n<p>So what does this do for us? Well, since we know these two things, we automatically know:<\/p>\n<div class=\"examplesentence\" style=\"font-size: 105%;\">\\[\\lim\\limits_{x \\to 0} \\frac{\\sin x}{x} = 1\\]<\/div>\n<p>\n&nbsp;<\/p>\n<ul>\n<li style=\"margin-bottom: 12px\">\\(\\frac{\\text{sin }x}{x}\\) is between \\(\\text{cos }x\\) and 1 around \\(x = 0\\)<\/li>\n<li style=\"text-align: left\"><math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>cos<\/mi><mo>&#x2061;<\/mo><mi>x<\/mi><mo>=<\/mo><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><mn>1<\/mn><mo>=<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/li>\n<\/ul>\n<p>\\(\\frac{\\text{sin }x}{x}\\) is squeezed between \\(\\text{cos }x\\) and 1 around \\(x=0\\). Since \\(\\text{cos }x\\) and 1 both approach 1 as \\(x\\) approaches 0, then \\(\\frac{\\text{sin }x}{x}\\) must as well. It\u2019s sort of an indirect way of finding a limit and it\u2019s easy to see why names like squeeze, pinch, sandwich, and police are used to describe the theorem.<\/p>\n<p>Now for the formal statement:<\/p>\n<p>If <math display=\"inline\"><mrow><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>&#x2264;<\/mo><mi>g<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>&#x2264;<\/mo><mi>h<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><\/math> and <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mi>a<\/mi><\/mrow><\/munder><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>=<\/mo><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mi>a<\/mi><\/mrow><\/munder><mi>h<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>=<\/mo><mi>L<\/mi><\/mrow><\/math>, then <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mi>a<\/mi><\/mrow><\/munder><mi>g<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><mo>=<\/mo><mi>L<\/mi><\/mrow><\/math>.<\/p>\n<h2><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h2>\n<p>\nUsing the theorem often involves more of a proof-like approach than a method and there are often many ways to use it to prove the value of a limit. For example, with the previous limit, we didn\u2019t need to use the functions \\(\\text{cos }x\\) and 1 as upper and lower bounds. We could have used \\(x^2+1\\) and \\(-x^2+1\\) since they both approach 1 as \\(x\\) approaches 0 and \\(\\frac{\\text{sin }x}{x}\\) is between them:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-72562\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-9.png\" alt=\"\" width=\"963\" height=\"492\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-9.png 963w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-9-300x153.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-9-768x392.png 768w\" sizes=\"auto, (max-width: 963px) 100vw, 963px\" \/><\/p>\n<p>Let\u2019s use the squeeze theorem to find <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&#x2061;<\/mo><mi>cos<\/mi><mo>(<\/mo><mfrac><mn>1<\/mn><mi>x<\/mi><\/mfrac><mo>)<\/mo><\/mrow><\/math>. The table and graph show the function oscillates around the \\(x\\)-axis, but the limit is 0:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-72550\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-1.png\" alt=\"\" width=\"1881\" height=\"489\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-1.png 1881w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-1-300x78.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-1-1024x266.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-1-768x200.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-1-1536x399.png 1536w\" sizes=\"auto, (max-width: 1881px) 100vw, 1881px\" \/><\/p>\n<p>Using substitution and algebra is fruitless, so we\u2019ll begin with the \\(\\text{cos }(\\frac{1}{x})\\) piece, because that \\(x\\) in the denominator is what\u2019s really complicating things.<\/p>\n<p>Based on our knowledge of the cosine function, we know \\(-1\u2264cos (\\frac{1}{x}) \u22641\\).<\/p>\n<p>Now multiply all three parts of the sentence by \\(x^2\\), because the result will be the original function:<\/p>\n<div class=\"examplesentence\">\\(-x^2\u2264x^2 \\text{cos }\\frac{1}{x}\u2264 x^2\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis tells us how to squeeze the function: Put it between \\(-x^2\\) and \\(x^2\\). Let\u2019s take a look.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-72565\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-11.png\" alt=\"\" width=\"1028\" height=\"556\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-11.png 1028w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-11-300x162.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-11-1024x554.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/05\/squeeze-theorem-11-768x415.png 768w\" sizes=\"auto, (max-width: 1028px) 100vw, 1028px\" \/><\/p>\n<p>According to the theorem, we know that <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><mo>&#x2212;<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>0<\/mn><mo>=<\/mo><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><\/math>. Now, because of the squeeze theorem, we know that <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&#x2061;<\/mo><mi>cos<\/mi><mo>(<\/mo><mfrac><mn>1<\/mn><mi>x<\/mi><\/mfrac><mo>)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><\/math>.<\/p>\n<h2><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h2>\n<p>\nSuppose \\(f(t)=-\\frac{2}{3}t^3+t^2+\\frac{1}{3}\\) and \\(h(t)=\\text{cos }\\frac{t\\pi}{2}\\). If \\(f(t)\u2264g(t)\u2264h(t)\\), use the squeeze theorem to show that <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>t<\/mi><mo>&#x2192;<\/mo><mn>2<\/mn><\/mrow><\/munder><mi>g<\/mi><mo>(<\/mo><mi>t<\/mi><mo>)<\/mo><mo>=<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/math>.<\/p>\n<p>Now, we don\u2019t know what \\(g(t)\\) is, but with the squeeze theorem we can prove this limit. All we need to show is the limits of \\(f(t)\\) and \\(h(t)\\) are the same as \\(x\\) approaches 2. Since we know that \\(g(t)\\) is pinched in between them, if those limits match, we\u2019ll have our limit.<\/p>\n<p>Let&#8217;s start by looking at <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>t<\/mi><mo>&#x2192;<\/mo><mn>2<\/mn><\/mrow><\/munder><mi>f<\/mi><mo>(<\/mo><mi>t<\/mi><mo>)<\/mo><\/mrow><\/math>. We&#8217;re going to plug in a 2 anywhere we see a \\(t\\) for this function.<\/p>\n<div class=\"examplesentence\"><math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>t<\/mi><mo>&#x2192;<\/mo><mn>2<\/mn><\/mrow><\/munder><mi>f<\/mi><mo>(<\/mo><mi>t<\/mi><mo>)<\/mo><\/mrow><\/math><math display=\"inline\"><mrow><mo>=<\/mo><mo>&#x2212;<\/mo><mfrac><mn>2<\/mn><mn>3<\/mn><\/mfrac><mo>(<\/mo><mn>2<\/mn><mo>)<\/mo><msup><mn>3<\/mn><mn>1<\/mn><\/msup><mo>+<\/mo><mo>(<\/mo><mn>2<\/mn><mo>)<\/mo><msup><mn>2<\/mn><mn>1<\/mn><\/msup><mo>+<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><\/math><math display=\"inline\"><mrow><mo>=<\/mo><mo>&#x2212;<\/mo><mfrac><mn>16<\/mn><mn>3<\/mn><\/mfrac><mo>+<\/mo><mn>4<\/mn><mo>+<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><\/math><math display=\"inline\"><mrow><mo>=<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/div>\n<p>\n&nbsp;<br \/>\nThen, we&#8217;re going to look at \\(h(t)\\). Again, we&#8217;re going to plug in a 2 anywhere we see a \\(t\\).<\/p>\n<div class=\"examplesentence\"><math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>t<\/mi><mo>&#x2192;<\/mo><mn>2<\/mn><\/mrow><\/munder><mi>h<\/mi><mo>(<\/mo><mi>t<\/mi><mo>)<\/mo><mo>=<\/mo><mi>cos<\/mi><mo>(<\/mo><mfrac><mrow><mn>2<\/mn><mi>&#x03C0;<\/mi><\/mrow><mn>2<\/mn><\/mfrac><mo>)<\/mo><\/mrow><\/math><math display=\"inline\"><mrow><mo>=<\/mo><mi>cos<\/mi><mo>(<\/mo><mi>&#x03C0;<\/mi><mo>)<\/mo><mo>=<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/div>\n<p>\n&nbsp;<br \/>\nSince the limits of \\(f(t)\\) and \\(h(t)\\) are both \u22121 and they are squeezing \\(g(t)\\), we know that <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>t<\/mi><mo>&#x2192;<\/mo><mn>2<\/mn><\/mrow><\/munder><mi>g<\/mi><mo>(<\/mo><mi>t<\/mi><mo>)<\/mo><mo>=<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/math>.<\/p>\n<h2><span id=\"Example_4\" class=\"m-toc-anchor\"><\/span>Example #4<\/h2>\n<p>\nLet\u2019s try one more!<\/p>\n<p>If \\(-\\frac{x^2}{4}-\\frac{x}{2}\u2264f(x)\u2264\\sqrt{-\\frac{1}{9x}}\\), use the squeeze theorem to find \\(\\lim_{x \\rightarrow -1}f(x)\\) .<\/p>\n<div class=\"examplesentence\"><math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>x<\/mi><mo>&#x2192;<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/munder><mo>&#x2212;<\/mo><mfrac><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mn>4<\/mn><\/mfrac><mo>&#x2212;<\/mo><mfrac><mi>x<\/mi><mn>2<\/mn><\/mfrac><mo>=<\/mo><mo>&#x2212;<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><\/math><\/p>\n<p style=\"margin-bottom: 0em;\"><math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>t<\/mi><mo>&#x2192;<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/munder><msqrt><mrow><mo>&#x2212;<\/mo><mfrac><mn>1<\/mn><mrow><mn>9<\/mn><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><\/msqrt><mo>=<\/mo><msqrt><mfrac><mn>1<\/mn><mn>9<\/mn><\/mfrac><\/msqrt><mo>=<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><\/math><\/p>\n<\/div>\n<p>\n&nbsp;<br \/>\nIn this case, \\(f(x)\\) is between the two functions, but it\u2019s not being squeezed. Since their limits as \\(x\\) approaches \u22121 do not match, the best we can say is that <math display=\"inline\"><mrow><munder><mo>lim<\/mo><mrow><mi>t<\/mi><mo>&#x2192;<\/mo><mo>&#x2212;<\/mo><mn>1<\/mn><\/mrow><\/munder><mi>f<\/mi><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><\/math> is between \\(\\frac{1}{4}\\) and \\(\\frac{1}{3}\\).<\/p>\n<p>Thanks for squeezing a little time out of your busy schedule to explore this theorem! I hope this video helped you understand when it\u2019s needed and how it works!<\/p>\n<p>See you next time!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Squeeze_Theorem_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Squeeze Theorem Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich of the following sets of functions \\(f(x)\\) and \\(h(x)\\) (denoted with red lines) would be most appropriate to use to apply the squeeze theorem and find \\(g(x)\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Squeeze-theorem-graph-example-1.svg\" alt=\"Graph showing two functions: a blue wavy curve with multiple peaks and valleys, and a red parabola-shaped curve opening upwards, both on a grid with labeled axes.\" width=\"973\" height=\"731\" class=\"aligncenter size-full wp-image-275134\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-1-2\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Squeeze-theorem-graph-example-2.svg\" alt=\"A graph showing a blue quintic function with multiple peaks and troughs, and a red upward-opening parabola centered at the origin.\" width=\"973\" height=\"731\" class=\"aligncenter size-full wp-image-275137\"  role=\"img\" \/><\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Squeeze-theorem-graph-example-3.svg\" alt=\"A graph showing two functions: a blue curve with multiple peaks and troughs, and a red curve with two branches opening upward and downward from the origin.\" width=\"973\" height=\"731\" class=\"aligncenter size-full wp-image-275140\"  role=\"img\" \/><\/div><div class=\"PQ\"  id=\"PQ-1-4\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Squeeze-theorem-graph-example-4.svg\" alt=\"A blue oscillating graph is plotted on a grid with two horizontal red lines above and below the x-axis, and black arrows indicating direction on the axes.\" width=\"973\" height=\"731\" class=\"aligncenter size-full wp-image-275143\"  role=\"img\" \/><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The squeeze theorem states that if \\(f(x)\\), \\(g(x)\\), and \\(h(x)\\) are functions such that \\(f(x)\\le g(x)\\le h(x)\\) for \\(x\\) near \\(a\\), and \\(\\lim_{x\\rightarrow a} f(x)=\\lim_{x\\rightarrow a} h(x)=L\\), then \\(\\lim_{x\\rightarrow a} g(x)=L\\).<\/p>\n<p>In other words, one way to find the limit of a function \\(g(x)\\) as \\(x\\) approaches some point \\(a\\) is by finding two functions that enclose \\(g(x)\\). If \\(f(x)\\) and \\(h(x)\\) come to the same limit \\(L\\) as \\(x\\) approaches \\(a\\), then \\(g(x)\\) must also approach \\(L\\).<\/p>\n<p>Notice the two conditions of the squeeze theorem:<\/p>\n<ol>\n<li style=\"margin-bottom: 10px\">The functions \\(f(x)\\) and \\(h(x)\\) must stay above and below \\(g(x)\\), respectively, for \\(x\\) near the point of interest.<\/li>\n<li>The functions \\(f(x)\\) and \\(h(x)\\) must approach the same limit value.<\/li>\n<\/ol>\n<p>In Choice A, the two red functions both fall below \\(g(x)\\) in the region near \\(x=0\\). Since \\(g(x)\\) is not sandwiched between the two functions near the point where the limit is taken, these functions cannot be used with the squeeze theorem.<\/p>\n<p>Choice B shows two red functions that lie above and below \\(g(x)\\) near \\(x=0\\). However, they approach different \\(y\\)-values as \\(x\\) approaches zero. The top function approaches \\(y=1\\), while the bottom function approaches \\(y=-1\\). Because these limits are different, the squeeze theorem does not apply, and the exact value of \\(\\lim_{x\\to 0} g(x)\\) cannot be determined from these functions.<\/p>\n<p>Choice C shows two red functions that lie entirely above and entirely below \\(g(x)\\), respectively, near \\(x=0\\). In addition, as \\(x\\) approaches zero, both of these functions approach the same value, \\(y=0\\). Therefore, the squeeze theorem applies, and it follows that \\(\\lim_{x\\to 0} g(x)=0\\).<\/p>\n<p>Choice D shows two red functions that lie above and below \\(g(x)\\) near \\(x=0\\), but they approach different values. The top function approaches 2, while the bottom function approaches \\(y=-1\\). Since the two bounding functions do not approach the same limit, the squeeze theorem cannot be used to determine the exact value of \\(\\lim_{x\\to 0} g(x)\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nGiven the information below, use the squeeze theorem to determine \\(\\lim_{x \\to 3}g(x)\\).<\/p>\n<ul>\n<li style=\"margin-bottom: 12px\">\\(f(x)=-2x^2+12x-18\\)<\/li>\n<li style=\"margin-bottom: 12px\">\\(h(x)=\\frac{1}{3}x^2-2x+3\\)<\/li>\n<li>\\(f(x)\\le g(x)\\le h(x)\\)<\/li>\n<\/ul>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\u22122<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">0<\/div><div class=\"PQ\"  id=\"PQ-2-3\">3<\/div><div class=\"PQ\"  id=\"PQ-2-4\">The exact value of <span style=\"font-size: 95%\">\\(\\lim_{x \\to 3}g(x)\\)<\/span> cannot be determined based on the information given.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Even though the problem doesn\u2019t explicitly state the function \\(g(x)\\), the squeeze theorem can be used to determine \\(\\lim_{x \\to 3} g(x)\\), as long as the two conditions of the theorem are met.<\/p>\n<p>The squeeze theorem states that if \\(f(x)\\), \\(g(x)\\), and \\(h(x)\\) are functions such that \\(f(x)\\le g(x)\\le h(x)\\) for \\(x\\) near \\(a\\), and \\(\\lim_{x\\to a} f(x)=\\lim_{x\\to a} h(x)=L\\), then \\(\\lim_{x\\to a} g(x)=L\\).<\/p>\n<p>In this problem, the first condition, \\(f(x)\\le g(x)\\le h(x)\\), is given as true. This means that to find the limit of \\(g(x)\\) as \\(x\\) approaches 3, we check whether \\(f(x)\\) and \\(h(x)\\) approach the same value as \\(x\\) approaches 3.<\/p>\n<p>Because \\(f(x)\\) and \\(h(x)\\) are polynomials, their limits can be found by direct substitution.<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(f(x)=-2x^2+12x-18\\)<br \/>\n\\(f(3)=-2(3)^2+12(3)-18\\)<br \/>\n\\(=-2(9)+36-18\\)<br \/>\n\\(=-18+36-18\\)<br \/>\n\\(=0\\)\n<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(h(x)=\\frac{1}{3}x^2-2x+3\\)<br \/>\n\\(h(3)=\\frac{1}{3}(3)^2-2(3)+3\\)<br \/>\n\\(=\\frac{1}{3}(9)-6+3\\)<br \/>\n\\(=3-6+3\\)<br \/>\n\\(=0\\)\n<\/p>\n<p>So \\(\\lim_{x \\to 3} f(x)=\\lim_{x \\to 3} h(x)=0\\).<\/p>\n<p>Since both bounding functions approach the same value, the squeeze theorem applies, and \\(\\lim_{x \\to 3} g(x)=0\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nLet \\(a(x)=\\frac{1}{4}x^2+x+3\\) and \\(c(x)=x^2+4x+6\\).<\/p>\n<p>Apply the squeeze theorem to determine the value of \\(\\lim_{x \\to -2}b(x)\\), given that \\(a(x)\\le b(x)\\le c(x)\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">0<\/div><div class=\"PQ\"  id=\"PQ-3-2\">1<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-3\">2<\/div><div class=\"PQ\"  id=\"PQ-3-4\">3<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>All that is needed to apply the squeeze theorem here is determining whether \\(a\\) and \\(c\\) have the same limit as \\(x\\) approaches \u22122. Both of those limits can be found by plugging in \\(x=-2\\).<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(a(x)=\\frac{1}{4}(-2)^2+(-2)+3\\)<br \/>\n\\(=\\frac{1}{4}(4)-2+3\\)<br \/>\n\\(=1-2+3\\)<br \/>\n\\(=2\\)\n<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(c(x)=(-2)^2+4(-2)+6\\)<br \/>\n\\(=4-8+6\\)<br \/>\n\\(=2\\)\n<\/p>\n<p>Both \\(a\\) and \\(c\\) have a limit of 2 as \\(x\\) approaches \u22122. Because the function \\(b(x)\\) lies between \\(a(x)\\) and \\(c(x)\\), the squeeze theorem dictates that \\(\\lim_{x \\to -2}b(x)\\) is also 2.\u2003<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nCarlos is modeling daily average gas prices for the month and determines that the price of gas in dollars for day \\(x\\) can be represented by some function \\(g(x)\\) which lies between the curves \\(f(x)=\\sqrt{x+1}\\) and \\(h(x)=\\frac{1}{6}x+\\frac{5}{3}\\).<\/p>\n<p>Can the squeeze theorem be applied to determine <span style=\"font-size: 95%\">\\(\\lim_{x \\to 8}g(x)\\)<\/span>? If so, what is the value of the limit?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">No<\/div><div class=\"PQ\"  id=\"PQ-4-2\">Yes: \\(g(x)=\\frac{3}{2}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-3\">Yes: \\(g(x)=3\\)<\/div><div class=\"PQ\"  id=\"PQ-4-4\">Yes: \\(g(x)=\\frac{11}{3}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The first condition of the squeeze theorem is met. The only thing necessary to check is if \\(f\\) and \\(h\\) come to the same limit as \\(x\\) approaches 8. This can be checked by substituting \\(x=8\\) into each function.<\/p>\n<p style=\"text-align: center;\">\\(f(x)=\\sqrt{8+1}=\\sqrt 9=\\pm 3\\)<\/p>\n<p>Since we are talking about gas prices, only use the positive value for this limit.<\/p>\n<p class=\"longmath\" style=\"text-align: center;\">\\(h(x)=\\dfrac{1}{6}(8)+\\dfrac{5}{3}=\\dfrac{8}{6}+\\dfrac{5}{3}=\\dfrac{4}{3}+\\dfrac{5}{3}=\\dfrac{9}{3}=3\\)<\/p>\n<p>It is now clear that \\(f\\) and \\(h\\) do come to the same limit as \\(x\\) approaches 8. Therefore, the squeeze theorem can be applied to find the value of \\(g(x)\\). That limit is the same as \\(f(x)\\) and \\(h(x)\\), which is 3.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nEloise is working to determine the effectiveness of her company\u2019s new marketing strategies by plotting revenue by day, where \\(x\\) represents the days since the new strategies were implemented. She determines that the revenue function \\(r(x)\\) is bounded below by the function \\(p(x)=30x+5{,}000\\) and bounded above by the function \\(q(x)=\\frac{3}{4}x^2+5{,}300\\).<\/p>\n<p>What is the value of \\(\\lim_{x \\to 20}r(x)\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">\\(\\lim_{x \\to 20}r(x)=5{,}600\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\(\\lim_{x \\to 20}r(x)=5{,}900\\)<\/div><div class=\"PQ\"  id=\"PQ-5-3\">\\(\\lim_{x \\to 20}r(x)=6{,}300\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\">\\(\\lim_{x \\to 20}r(x)=6{,}550\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To use the squeeze theorem, \\(r(x)\\) must lie between \\(p(x)\\) and \\(q(x)\\), and \\(p\\) and \\(q\\) must have the same limit as \\(x\\) approaches 20.<\/p>\n<p>In this problem, it is given that \\(r\\) is between \\(p\\) and \\(q\\) when the problem states that \\(r\\) is bounded below by \\(p\\) and bounded above by \\(q\\). This means the only remaining condition is checking if \\(p\\) and \\(q\\) have the same limit as \\(x\\) approaches 20. This can be done by substituting \\(x=20\\) into both functions.<\/p>\n<p class=\"longmath\" style=\"text-align: center;\">\\(p(x)=30(20)+5{,}000=600+5{,}000=5{,}600\\)<\/p>\n<p class=\"longmath\" style=\"text-align: center;\">\\(q(x)=\\frac{3}{4}(20)^2+5{,}300=\\frac{3}{4}(400)+5{,}300=300+5{,}300=5{,}600\\)<\/p>\n<p>Because \\(p\\) and \\(q\\) have the same limit as \\(x\\) approaches 20, and because \\(r\\) lies between them, the squeeze theorem dictates that \\(\\lim_{x \\to 20}r(x)=5{,}600\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/calculus\/\">Return to Calculus Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Calculus Videos<\/p>\n","protected":false},"author":1,"featured_media":100771,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-72544","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-calculus-videos","7":"page_category-video-pages-for-study-course-sidebar-ad","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO Pro 4.9.10 - aioseo.com -->\n\t<meta name=\"description\" content=\"The squeeze theorem uses multiple functions on a graph to determine the limits of another as it approaches a value. 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