{"id":2066,"date":"2025-08-29T14:12:26","date_gmt":"2025-08-29T14:12:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=2066"},"modified":"2025-08-29T14:14:49","modified_gmt":"2025-08-29T14:14:49","slug":"numerical-and-improper-integration-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/numerical-and-improper-integration-background-youll-need-3\/","title":{"raw":"Numerical and Improper Integration: Background You'll Need 3","rendered":"Numerical and Improper Integration: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Evaluate one-sided limits<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>One-Sided Limits<\/h2>\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, lets look at the function [latex]g(x)=\\frac{|x-2|}{(x-2)}[\/latex],<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202849\/CNX_Calc_Figure_02_02_001.jpg\" alt=\"&quot;Three 2 and x= -1 for x &lt; 2. There are open circles at both endpoints (2, 1) and (-2, 1). The third is h(x) = 1 \/ (x-2)^2, in which the function curves asymptotically towards y=0 and x=2 in quadrants one and two.&quot; width=&quot;975&quot; height=&quot;434&quot;\" width=\"975\" height=\"434\" \/> Figure 1. These graphs show the behavior of three different functions around [latex]x=2[\/latex].[\/caption]As we pick values of [latex]x[\/latex] close to [latex]2[\/latex], [latex]g(x)[\/latex] does not approach a single value, so the limit as [latex]x[\/latex] approaches [latex]2[\/latex] does not exist\u2014that is, [latex]\\underset{x\\to 2}{\\lim}g(x)[\/latex] DNE.\r\n\r\nHowever, this statement alone does not give us a complete picture of the behavior of the function around the [latex]x[\/latex]-value [latex]2[\/latex]. To provide a more accurate description, we introduce the idea of a <strong>one-sided limit<\/strong>.\r\n\r\nFor all values to the left of [latex]2[\/latex] (or <em>the negative side of<\/em> [latex]2[\/latex]), [latex]g(x)=-1[\/latex]. Thus, as [latex]x[\/latex] approaches [latex]2[\/latex] from the left, [latex]g(x)[\/latex] approaches [latex]\u22121[\/latex].\r\n\r\nMathematically, we say that the limit as [latex]x[\/latex] approaches [latex]2[\/latex] from the left is [latex]\u22121[\/latex]. Symbolically, we express this idea as\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<p id=\"fs-id1170571569214\">Similarly, as [latex]x[\/latex] approaches [latex]2[\/latex] from the right (or <em>from the positive side<\/em>), [latex]g(x)[\/latex] approaches [latex]1[\/latex]. 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<section class=\"textbox keyTakeaway\">\r\n<h3>one-sided limits<\/h3>\r\n<p id=\"fs-id1170572307699\">One-sided limits are limits approached from one direction\u2014either from the left or the right.<\/p>\r\n\r\n<ul>\r\n \t<li><strong>Left-Sided Limit:<\/strong> For a function [latex]f(x)[\/latex] on an interval ending at [latex]a[\/latex], if [latex]f(x)[\/latex] approaches a specific value [latex]L[\/latex] as the values of [latex]x[\/latex] approaches [latex]a[\/latex] from the left ([latex]x &lt; a[\/latex]), we denote this limit as:<center>[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex]<\/center><\/li>\r\n \t<li><strong>Right-Sided Limit: <\/strong>For a function [latex]f(x)[\/latex] on an interval ending at [latex]a[\/latex], if [latex]f(x)[\/latex] approaches a specific value [latex]L[\/latex] as the values of [latex]x[\/latex] approaches [latex]a[\/latex] from the right\u00a0 ([latex]x &gt; a[\/latex]), we express this limit as:<center>[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/center><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">\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 [latex]2[\/latex], we use [latex]f(x)=x+1[\/latex] and for values of [latex]x[\/latex] greater than [latex]2[\/latex], 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>[latex]1.9[\/latex]<\/td>\r\n<td>[latex]2.9[\/latex]<\/td>\r\n<td rowspan=\"5\"><\/td>\r\n<td>[latex]2.1[\/latex]<\/td>\r\n<td>[latex]0.41[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]1.99[\/latex]<\/td>\r\n<td>[latex]2.99[\/latex]<\/td>\r\n<td>[latex]2.01[\/latex]<\/td>\r\n<td>[latex]0.0401[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]1.999[\/latex]<\/td>\r\n<td>[latex]2.999[\/latex]<\/td>\r\n<td>[latex]2.001[\/latex]<\/td>\r\n<td>[latex]0.004001[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]1.9999[\/latex]<\/td>\r\n<td>[latex]2.9999[\/latex]<\/td>\r\n<td>[latex]2.0001[\/latex]<\/td>\r\n<td>[latex]0.00040001[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]1.99999[\/latex]<\/td>\r\n<td>[latex]2.99999[\/latex]<\/td>\r\n<td>[latex]2.00001[\/latex]<\/td>\r\n<td>[latex]0.0000400001[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572233834\">Based on this table, we can conclude that<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to 2^-}{\\lim}f(x)=3[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 2^+}{\\lim}f(x)=0[\/latex].<\/li>\r\n<\/ol>\r\nTherefore, 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.\r\n\r\n[caption id=\"\" align=\"alignnone\" 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=\"&quot;The\" width=\"487\" height=\"431\" \/> A discontinuous piecewise function with an open circle and a closed circle at transition points[\/caption]\r\n\r\nWatch the following video to see the worked solution to this example.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.You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction576to688_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window). [\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Two-Sided Limits<\/h2>\r\nA two-sided limit at a point exists only if the one-sided limits from both the left and the right converge to the same value. If there's a discrepancy between the left and the right limits, the two-sided limit at that point does not exist.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>two-sided limits<\/h3>\r\nFor a function [latex]f(x)[\/latex], defined over an interval including [latex]a[\/latex] (except possibly at [latex]a[\/latex] itself), we say the two-sided limit exists as [latex]x[\/latex] approaches [latex]a[\/latex] and equals [latex]L[\/latex] if, and only if, both one-sided limits as [latex]x[\/latex] approaches [latex]a[\/latex] also equals [latex]L[\/latex].\r\n\r\n<center>[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex], if and only if [latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/center><\/section><section class=\"textbox tryIt\">[ohm_question hide_question_number=1]218961[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_number=1]284309[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Evaluate one-sided limits<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>One-Sided Limits<\/h2>\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, lets look at the function [latex]g(x)=\\frac{|x-2|}{(x-2)}[\/latex],<\/p>\n<figure style=\"width: 975px\" 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\/11202849\/CNX_Calc_Figure_02_02_001.jpg\" alt=\"&quot;Three 2 and x= -1 for x &lt; 2. There are open circles at both endpoints (2, 1) and (-2, 1). The third is h(x) = 1 \/ (x-2)^2, in which the function curves asymptotically towards y=0 and x=2 in quadrants one and two.&quot; width=&quot;975&quot; height=&quot;434&quot;\" width=\"975\" height=\"434\" \/><figcaption class=\"wp-caption-text\">Figure 1. These graphs show the behavior of three different functions around [latex]x=2[\/latex].<\/figcaption><\/figure>\n<p>As we pick values of [latex]x[\/latex] close to [latex]2[\/latex], [latex]g(x)[\/latex] does not approach a single value, so the limit as [latex]x[\/latex] approaches [latex]2[\/latex] does not exist\u2014that is, [latex]\\underset{x\\to 2}{\\lim}g(x)[\/latex] DNE.<\/p>\n<p>However, this statement alone does not give us a complete picture of the behavior of the function around the [latex]x[\/latex]-value [latex]2[\/latex]. To provide a more accurate description, we introduce the idea of a <strong>one-sided limit<\/strong>.<\/p>\n<p>For all values to the left of [latex]2[\/latex] (or <em>the negative side of<\/em> [latex]2[\/latex]), [latex]g(x)=-1[\/latex]. Thus, as [latex]x[\/latex] approaches [latex]2[\/latex] from the left, [latex]g(x)[\/latex] approaches [latex]\u22121[\/latex].<\/p>\n<p>Mathematically, we say that the limit as [latex]x[\/latex] approaches [latex]2[\/latex] from the left is [latex]\u22121[\/latex]. 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 id=\"fs-id1170571569214\">Similarly, as [latex]x[\/latex] approaches [latex]2[\/latex] from the right (or <em>from the positive side<\/em>), [latex]g(x)[\/latex] approaches [latex]1[\/latex]. 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<section class=\"textbox keyTakeaway\">\n<h3>one-sided limits<\/h3>\n<p id=\"fs-id1170572307699\">One-sided limits are limits approached from one direction\u2014either from the left or the right.<\/p>\n<ul>\n<li><strong>Left-Sided Limit:<\/strong> For a function [latex]f(x)[\/latex] on an interval ending at [latex]a[\/latex], if [latex]f(x)[\/latex] approaches a specific value [latex]L[\/latex] as the values of [latex]x[\/latex] approaches [latex]a[\/latex] from the left ([latex]x < a[\/latex]), we denote this limit as:\n\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex]<\/div>\n<\/li>\n<li><strong>Right-Sided Limit: <\/strong>For a function [latex]f(x)[\/latex] on an interval ending at [latex]a[\/latex], if [latex]f(x)[\/latex] approaches a specific value [latex]L[\/latex] as the values of [latex]x[\/latex] approaches [latex]a[\/latex] from the right\u00a0 ([latex]x > a[\/latex]), we express this limit as:\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572307130\">Show Solution<\/button><\/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 [latex]2[\/latex], we use [latex]f(x)=x+1[\/latex] and for values of [latex]x[\/latex] greater than [latex]2[\/latex], 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>[latex]1.9[\/latex]<\/td>\n<td>[latex]2.9[\/latex]<\/td>\n<td rowspan=\"5\"><\/td>\n<td>[latex]2.1[\/latex]<\/td>\n<td>[latex]0.41[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]1.99[\/latex]<\/td>\n<td>[latex]2.99[\/latex]<\/td>\n<td>[latex]2.01[\/latex]<\/td>\n<td>[latex]0.0401[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]1.999[\/latex]<\/td>\n<td>[latex]2.999[\/latex]<\/td>\n<td>[latex]2.001[\/latex]<\/td>\n<td>[latex]0.004001[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]1.9999[\/latex]<\/td>\n<td>[latex]2.9999[\/latex]<\/td>\n<td>[latex]2.0001[\/latex]<\/td>\n<td>[latex]0.00040001[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]1.99999[\/latex]<\/td>\n<td>[latex]2.99999[\/latex]<\/td>\n<td>[latex]2.00001[\/latex]<\/td>\n<td>[latex]0.0000400001[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572233834\">Based on this table, we can conclude that<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 2^-}{\\lim}f(x)=3[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2^+}{\\lim}f(x)=0[\/latex].<\/li>\n<\/ol>\n<p>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<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><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=\"&quot;The\" width=\"487\" height=\"431\" \/><figcaption class=\"wp-caption-text\">A discontinuous piecewise function with an open circle and a closed circle at transition points<\/figcaption><\/figure>\n<p>Watch the following video to see the worked solution to this example.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.You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction576to688_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window). <\/div>\n<\/div>\n<\/section>\n<h2>Two-Sided Limits<\/h2>\n<p>A two-sided limit at a point exists only if the one-sided limits from both the left and the right converge to the same value. If there&#8217;s a discrepancy between the left and the right limits, the two-sided limit at that point does not exist.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>two-sided limits<\/h3>\n<p>For a function [latex]f(x)[\/latex], defined over an interval including [latex]a[\/latex] (except possibly at [latex]a[\/latex] itself), we say the two-sided limit exists as [latex]x[\/latex] approaches [latex]a[\/latex] and equals [latex]L[\/latex] if, and only if, both one-sided limits as [latex]x[\/latex] approaches [latex]a[\/latex] also equals [latex]L[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex], if and only if [latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm218961\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218961&theme=lumen&iframe_resize_id=ohm218961&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm284309\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=284309&theme=lumen&iframe_resize_id=ohm284309&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":667,"module-header":"- Select Header -","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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/2066"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":2,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/2066\/revisions"}],"predecessor-version":[{"id":2068,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/2066\/revisions\/2068"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/667"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/2066\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=2066"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=2066"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=2066"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=2066"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}