{"id":281,"date":"2021-02-04T00:47:33","date_gmt":"2021-02-04T00:47:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=281"},"modified":"2022-03-11T21:51:29","modified_gmt":"2022-03-11T21:51:29","slug":"one-sided-limits","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/one-sided-limits\/","title":{"raw":"One-Sided Limits","rendered":"One-Sided Limits"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define one-sided limits and provide examples<\/li>\r\n \t<li>Explain the relationship between one-sided and two-sided limits<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170572334719\" class=\"bc-section section\">\r\n<p id=\"fs-id1170572334724\">Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point. To see this, we now revisit the function [latex]g(x)=\\frac{|x-2|}{(x-2)}[\/latex] introduced at the beginning of the section. As we pick values of [latex]x[\/latex] close to 2, [latex]g(x)[\/latex] does not approach a single value, so the limit as [latex]x[\/latex] approaches 2 does not exist\u2014that is, [latex]\\underset{x\\to 2}{\\lim}g(x)[\/latex] DNE. However, this statement alone does not give us a complete picture of the behavior of the function around the [latex]x[\/latex]-value 2. To provide a more accurate description, we introduce the idea of a <strong>one-sided limit<\/strong>. For all values to the left of 2 (or <em>the negative side of<\/em> 2), [latex]g(x)=-1[\/latex]. Thus, as [latex]x[\/latex] approaches 2 from the left, [latex]g(x)[\/latex] approaches \u22121. Mathematically, we say that the limit as [latex]x[\/latex] approaches 2 from the left is \u22121. Symbolically, we express this idea as<\/p>\r\n\r\n<div id=\"fs-id1170571655354\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 2^-}{\\lim}g(x)=-1[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170571569214\">Similarly, as [latex]x[\/latex] approaches 2 from the right (or <em>from the positive side<\/em>), [latex]g(x)[\/latex] approaches 1. Symbolically, we express this idea as<\/p>\r\n\r\n<div id=\"fs-id1170571569241\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 2^+}{\\lim}g(x)=1[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572307691\">We can now present an informal definition of one-sided limits.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572307699\">We define two types of one-sided limits.<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1170572307707\"><strong>Limit from the left:<\/strong> Let [latex]f(x)[\/latex] be a function defined at all values in an open interval of the form [latex](c,a)[\/latex], and let [latex]L[\/latex] be a real number. If the values of the function [latex]f(x)[\/latex] approach the real number [latex]L[\/latex]\u00a0as the values of [latex]x[\/latex] (where [latex]x&lt;a[\/latex]) approach the number [latex]a[\/latex], then we say that [latex]L[\/latex]\u00a0is the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the left. Symbolically, we express this idea as<\/p>\r\n\r\n<div id=\"fs-id1170571531299\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170571655616\"><strong>Limit from the right:<\/strong> Let [latex]f(x)[\/latex] be a function defined at all values in an open interval of the form [latex](a,c)[\/latex], and let [latex]L[\/latex]\u00a0be a real number. If the values of the function [latex]f(x)[\/latex] approach the real number [latex]L[\/latex] as the values of [latex]x[\/latex] (where [latex]x&gt;a[\/latex]) approach the number [latex]a[\/latex], then we say that [latex]L[\/latex]\u00a0is the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the right. Symbolically, we express this idea as<\/p>\r\n\r\n<div id=\"fs-id1170572453163\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571614880\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating One-Sided Limits<\/h3>\r\n<p id=\"fs-id1170571614889\">For the function [latex]f(x)=\\begin{cases} x+1, &amp; \\text{ if } \\, x &lt; 2 \\\\ x^2-4, &amp; \\text{ if } \\, x \\ge 2 \\end{cases}[\/latex], evaluate each of the following limits.<\/p>\r\n\r\n<ol id=\"fs-id1170571596873\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to 2^-}{\\lim}f(x)[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 2^+}{\\lim}f(x)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572307130\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572307130\"]\r\n<p id=\"fs-id1170572307130\">We can use tables of functional values again. Observe that for values of [latex]x[\/latex] less than 2, we use [latex]f(x)=x+1[\/latex] and for values of [latex]x[\/latex] greater than 2, we use [latex]f(x)=x^2-4[\/latex].<\/p>\r\n\r\n<table id=\"fs-id1170572347185\" summary=\"Two tables side by side, each with two columns and six rows. The headers are the same, x and f(x) = x+1 in the first row. In the first table, the values in the first column under x are 1.9, 1.99, 1.999, 1.9999, and 1.99999. The values in the second column under the header are 2.9, 2.99, 2.999, 2.9999, and 2.99999. In the second column, the values in the first column under x are 2.1, 2.01, 2.001, 2.0001, and 2.00001. The values in the second column under the header are 0.41, 0.0401, 0.004001, 0.00040001, and 0.0000400001.\"><caption>Table of Functional Values for [latex]f(x)=\\begin{cases} x+1, &amp; \\text{ if } \\, x &lt; 2 \\\\ x^2-4, &amp; \\text{ if } \\, x \\ge 2 \\end{cases}[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)=x+1[\/latex]<\/th>\r\n<th><\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)=x^2-4[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>1.9<\/td>\r\n<td>2.9<\/td>\r\n<td rowspan=\"5\"><\/td>\r\n<td>2.1<\/td>\r\n<td>0.41<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1.99<\/td>\r\n<td>2.99<\/td>\r\n<td>2.01<\/td>\r\n<td>0.0401<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1.999<\/td>\r\n<td>2.999<\/td>\r\n<td>2.001<\/td>\r\n<td>0.004001<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1.9999<\/td>\r\n<td>2.9999<\/td>\r\n<td>2.0001<\/td>\r\n<td>0.00040001<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>1.99999<\/td>\r\n<td>2.99999<\/td>\r\n<td>2.00001<\/td>\r\n<td>0.0000400001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572233834\">Based on this table, we can conclude that a. [latex]\\underset{x\\to 2^-}{\\lim}f(x)=3[\/latex] and b. [latex]\\underset{x\\to 2^+}{\\lim}f(x)=0[\/latex]. Therefore, the (two-sided) limit of [latex]f(x)[\/latex] does not exist at [latex]x=2[\/latex]. Figure 7 shows a graph of [latex]f(x)[\/latex] and reinforces our conclusion about these limits.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202909\/CNX_Calc_Figure_02_02_010.jpg\" alt=\"The graph of the given piecewise function. The first piece is f(x) = x+1 if x &lt; 2. The second piece is x^2 \u2013 4 if x &gt;= 2. The first piece is a line with x intercept at (-1, 0) and y intercept at (0,1). There is an open circle at (2,3), where the endpoint would be. The second piece is the right half of a parabola opening upward. The vertex at (2,0) is a solid circle.\" width=\"487\" height=\"431\" \/> Figure 7. The graph of [latex]f(x)=\\begin{cases} x+1, &amp; \\text{ if } \\, x &lt; 2 \\\\ x^2-4, &amp; \\text{ if } \\, x \\ge 2 \\end{cases}[\/latex] has a break at [latex]x=2[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Evaluating One-Sided Limits[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qiHi41CfnFA?controls=0&amp;start=576&amp;end=688&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction576to688_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.2 The Limit of a Function\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170571612124\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571612132\">Use a table of functional values to estimate the following limits, if possible.<\/p>\r\n\r\n<ol id=\"fs-id1170571612135\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 2^+}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"228744\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"228744\"]\r\n<ol id=\"fs-id1170572452145\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Use [latex]x[\/latex]-values 1.9, 1.99, 1.999, 1.9999, 1.9999 to estimate [latex]\\underset{x\\to 2^-}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex].<\/li>\r\n \t<li>Use [latex]x[\/latex]-values 2.1, 2.01, 2.001, 2.0001, 2.00001 to estimate [latex]\\underset{x\\to 2^+}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex].\r\n(These tables are available from a previous Checkpoint problem.)<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572306438\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572306438\"]\r\n<p id=\"fs-id1170572306438\">a. [latex]\\underset{x\\to 2^-}{\\lim}\\frac{|x^2-4|}{x-2}=-4[\/latex]<\/p>\r\nb. [latex]\\underset{x\\to 2^+}{\\lim}\\frac{|x^2-4|}{x-2}=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]20381[\/ohm_question]\r\n\r\n<\/div>\r\nLet us now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. It seems clear that if the limit from the right and the limit from the left have a common value, then that common value is the limit of the function at that point. Similarly, if the limit from the left and the limit from the right take on different values, the limit of the function does not exist.\r\n<div class=\"textbox shaded\">\r\n<h3 class=\"os-subtitle\" style=\"text-align: center;\" data-type=\"title\"><span class=\"os-subtitle-label\">Relating One-Sided and Two-Sided Limits<\/span><\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572560622\">Let [latex]f(x)[\/latex]\u00a0be a function defined at all values in an open interval containing [latex]a[\/latex], with the possible exception of [latex]a[\/latex]\u00a0itself, and let [latex]L[\/latex]\u00a0be a real number. Then,<\/p>\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex], if and only if\u00a0[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex] and\u00a0[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]218961[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define one-sided limits and provide examples<\/li>\n<li>Explain the relationship between one-sided and two-sided limits<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170572334719\" class=\"bc-section section\">\n<p id=\"fs-id1170572334724\">Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point. To see this, we now revisit the function [latex]g(x)=\\frac{|x-2|}{(x-2)}[\/latex] introduced at the beginning of the section. As we pick values of [latex]x[\/latex] close to 2, [latex]g(x)[\/latex] does not approach a single value, so the limit as [latex]x[\/latex] approaches 2 does not exist\u2014that is, [latex]\\underset{x\\to 2}{\\lim}g(x)[\/latex] DNE. However, this statement alone does not give us a complete picture of the behavior of the function around the [latex]x[\/latex]-value 2. To provide a more accurate description, we introduce the idea of a <strong>one-sided limit<\/strong>. For all values to the left of 2 (or <em>the negative side of<\/em> 2), [latex]g(x)=-1[\/latex]. Thus, as [latex]x[\/latex] approaches 2 from the left, [latex]g(x)[\/latex] approaches \u22121. Mathematically, we say that the limit as [latex]x[\/latex] approaches 2 from the left is \u22121. Symbolically, we express this idea as<\/p>\n<div id=\"fs-id1170571655354\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 2^-}{\\lim}g(x)=-1[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170571569214\">Similarly, as [latex]x[\/latex] approaches 2 from the right (or <em>from the positive side<\/em>), [latex]g(x)[\/latex] approaches 1. Symbolically, we express this idea as<\/p>\n<div id=\"fs-id1170571569241\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 2^+}{\\lim}g(x)=1[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572307691\">We can now present an informal definition of one-sided limits.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1170572307699\">We define two types of one-sided limits.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572307707\"><strong>Limit from the left:<\/strong> Let [latex]f(x)[\/latex] be a function defined at all values in an open interval of the form [latex](c,a)[\/latex], and let [latex]L[\/latex] be a real number. If the values of the function [latex]f(x)[\/latex] approach the real number [latex]L[\/latex]\u00a0as the values of [latex]x[\/latex] (where [latex]x<a[\/latex]) approach the number [latex]a[\/latex], then we say that [latex]L[\/latex]\u00a0is the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the left. Symbolically, we express this idea as<\/p>\n<div id=\"fs-id1170571531299\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170571655616\"><strong>Limit from the right:<\/strong> Let [latex]f(x)[\/latex] be a function defined at all values in an open interval of the form [latex](a,c)[\/latex], and let [latex]L[\/latex]\u00a0be a real number. If the values of the function [latex]f(x)[\/latex] approach the real number [latex]L[\/latex] as the values of [latex]x[\/latex] (where [latex]x>a[\/latex]) approach the number [latex]a[\/latex], then we say that [latex]L[\/latex]\u00a0is the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the right. Symbolically, we express this idea as<\/p>\n<div id=\"fs-id1170572453163\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1170571614880\" class=\"textbook exercises\">\n<h3>Example: Evaluating One-Sided Limits<\/h3>\n<p id=\"fs-id1170571614889\">For the function [latex]f(x)=\\begin{cases} x+1, & \\text{ if } \\, x < 2 \\\\ x^2-4, & \\text{ if } \\, x \\ge 2 \\end{cases}[\/latex], evaluate each of the following limits.<\/p>\n<ol id=\"fs-id1170571596873\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 2^-}{\\lim}f(x)[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2^+}{\\lim}f(x)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572307130\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572307130\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572307130\">We can use tables of functional values again. Observe that for values of [latex]x[\/latex] less than 2, we use [latex]f(x)=x+1[\/latex] and for values of [latex]x[\/latex] greater than 2, we use [latex]f(x)=x^2-4[\/latex].<\/p>\n<table id=\"fs-id1170572347185\" summary=\"Two tables side by side, each with two columns and six rows. The headers are the same, x and f(x) = x+1 in the first row. In the first table, the values in the first column under x are 1.9, 1.99, 1.999, 1.9999, and 1.99999. The values in the second column under the header are 2.9, 2.99, 2.999, 2.9999, and 2.99999. In the second column, the values in the first column under x are 2.1, 2.01, 2.001, 2.0001, and 2.00001. The values in the second column under the header are 0.41, 0.0401, 0.004001, 0.00040001, and 0.0000400001.\">\n<caption>Table of Functional Values for [latex]f(x)=\\begin{cases} x+1, & \\text{ if } \\, x < 2 \\\\ x^2-4, & \\text{ if } \\, x \\ge 2 \\end{cases}[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)=x+1[\/latex]<\/th>\n<th><\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)=x^2-4[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>1.9<\/td>\n<td>2.9<\/td>\n<td rowspan=\"5\"><\/td>\n<td>2.1<\/td>\n<td>0.41<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1.99<\/td>\n<td>2.99<\/td>\n<td>2.01<\/td>\n<td>0.0401<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1.999<\/td>\n<td>2.999<\/td>\n<td>2.001<\/td>\n<td>0.004001<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1.9999<\/td>\n<td>2.9999<\/td>\n<td>2.0001<\/td>\n<td>0.00040001<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>1.99999<\/td>\n<td>2.99999<\/td>\n<td>2.00001<\/td>\n<td>0.0000400001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572233834\">Based on this table, we can conclude that a. [latex]\\underset{x\\to 2^-}{\\lim}f(x)=3[\/latex] and b. [latex]\\underset{x\\to 2^+}{\\lim}f(x)=0[\/latex]. Therefore, the (two-sided) limit of [latex]f(x)[\/latex] does not exist at [latex]x=2[\/latex]. Figure 7 shows a graph of [latex]f(x)[\/latex] and reinforces our conclusion about these limits.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202909\/CNX_Calc_Figure_02_02_010.jpg\" alt=\"The graph of the given piecewise function. The first piece is f(x) = x+1 if x &lt; 2. The second piece is x^2 \u2013 4 if x &gt;= 2. The first piece is a line with x intercept at (-1, 0) and y intercept at (0,1). There is an open circle at (2,3), where the endpoint would be. The second piece is the right half of a parabola opening upward. The vertex at (2,0) is a solid circle.\" width=\"487\" height=\"431\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7. The graph of [latex]f(x)=\\begin{cases} x+1, &amp; \\text{ if } \\, x &lt; 2 \\\\ x^2-4, &amp; \\text{ if } \\, x \\ge 2 \\end{cases}[\/latex] has a break at [latex]x=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Evaluating One-Sided Limits<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qiHi41CfnFA?controls=0&amp;start=576&amp;end=688&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction576to688_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.2 The Limit of a Function&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170571612124\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571612132\">Use a table of functional values to estimate the following limits, if possible.<\/p>\n<ol id=\"fs-id1170571612135\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2^+}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q228744\">Hint<\/span><\/p>\n<div id=\"q228744\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572452145\" style=\"list-style-type: lower-alpha;\">\n<li>Use [latex]x[\/latex]-values 1.9, 1.99, 1.999, 1.9999, 1.9999 to estimate [latex]\\underset{x\\to 2^-}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex].<\/li>\n<li>Use [latex]x[\/latex]-values 2.1, 2.01, 2.001, 2.0001, 2.00001 to estimate [latex]\\underset{x\\to 2^+}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex].<br \/>\n(These tables are available from a previous Checkpoint problem.)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572306438\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572306438\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572306438\">a. [latex]\\underset{x\\to 2^-}{\\lim}\\frac{|x^2-4|}{x-2}=-4[\/latex]<\/p>\n<p>b. [latex]\\underset{x\\to 2^+}{\\lim}\\frac{|x^2-4|}{x-2}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm20381\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20381&theme=oea&iframe_resize_id=ohm20381&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Let us now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. It seems clear that if the limit from the right and the limit from the left have a common value, then that common value is the limit of the function at that point. Similarly, if the limit from the left and the limit from the right take on different values, the limit of the function does not exist.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"os-subtitle\" style=\"text-align: center;\" data-type=\"title\"><span class=\"os-subtitle-label\">Relating One-Sided and Two-Sided Limits<\/span><\/h3>\n<hr \/>\n<p id=\"fs-id1170572560622\">Let [latex]f(x)[\/latex]\u00a0be a function defined at all values in an open interval containing [latex]a[\/latex], with the possible exception of [latex]a[\/latex]\u00a0itself, and let [latex]L[\/latex]\u00a0be a real number. Then,<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex], if and only if\u00a0[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex] and\u00a0[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm218961\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218961&theme=oea&iframe_resize_id=ohm218961&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-281\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>2.2 The Limit of a Function. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"2.2 The Limit of a Function\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-281","chapter","type-chapter","status-publish","hentry"],"part":28,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/281","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":29,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/281\/revisions"}],"predecessor-version":[{"id":4768,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/281\/revisions\/4768"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/28"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/281\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=281"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=281"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=281"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=281"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}