{"id":743,"date":"2025-06-20T17:09:27","date_gmt":"2025-06-20T17:09:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=743"},"modified":"2025-08-29T14:14:48","modified_gmt":"2025-08-29T14:14:48","slug":"numerical-and-improper-integration-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/numerical-and-improper-integration-background-youll-need-2\/","title":{"raw":"Numerical and Improper Integration: Background You'll Need 2","rendered":"Numerical and Improper Integration: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Evaluate limits at infinity<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Infinite Limits<\/h2>\r\nEvaluating limits, whether at a specific point or as we approach it from a particular direction, helps us understand how functions behave near that point. While some functions have limits that are finite numbers, others grow without bound\u2014these are cases of infinite limits.\r\n<div id=\"fs-id1170571611973\" class=\"bc-section section\">\r\n\r\nWe now turn our attention to [latex]h(x)=\\frac{1}{(x-2)^2}[\/latex] (Figure 1 part c).\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 [latex]x[\/latex] gets closer to [latex]2[\/latex], [latex]h(x)[\/latex] increases without limit. This unbounded growth means that as [latex]x[\/latex] approaches [latex]2[\/latex], [latex]h(x)[\/latex] heads towards positive infinity, which we denote as:\r\n<div id=\"fs-id1170571612232\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\lim}h(x)=+\\infty [\/latex]<\/div>\r\n<p id=\"fs-id1170571612271\">Infinite limits can be understood through the following general definitions.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>infinite limits<\/h3>\r\n<p id=\"fs-id1170571612290\"><strong>Infinite limits from the left:<\/strong> For a function [latex]f(x)[\/latex] within an interval that ends at [latex]a[\/latex], we say:<\/p>\r\n\r\n<ol id=\"fs-id1170571562562\">\r\n \t<li>The limit is [latex]+\u221e[\/latex] if [latex]f(x)[\/latex] increases without bound as [latex]x[\/latex] approaches [latex]a[\/latex] from the left.\r\n<center>[latex]\\underset{x\\to a^-}{\\lim}f(x)=+\\infty[\/latex].<\/center><\/li>\r\n \t<li>The limit is [latex]-\u221e[\/latex] if [latex]f(x)[\/latex] decreases without bound as [latex]x[\/latex] approaches [latex]a[\/latex] from the left.\r\n<center>[latex]\\underset{x\\to a^-}{\\lim}f(x)=\u2212\\infty[\/latex].<\/center><\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572346754\"><strong>Infinite limits from the right:<\/strong> For a function [latex]f(x)[\/latex] within an interval that ends at [latex]a[\/latex], we say:<\/p>\r\n\r\n<ol id=\"fs-id1170572346792\">\r\n \t<li>The limit is [latex]+\u221e[\/latex] if [latex]f(x)[\/latex] increases without bound as [latex]x[\/latex] approaches [latex]a[\/latex] from the right.\r\n<div id=\"fs-id1170572559800\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=+\\infty[\/latex].<\/div><\/li>\r\n \t<li>The limit is [latex]-\u221e[\/latex] if [latex]f(x)[\/latex] decreases without bound as [latex]x[\/latex] approaches [latex]a[\/latex] from the right.\r\n<div id=\"fs-id1170572512575\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=\u2212\\infty[\/latex].<\/div><\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170572512615\"><strong>Two-sided infinite limit: <\/strong>For a function [latex]f(x)[\/latex] defined at all points except at [latex]a[\/latex]:<\/p>\r\n\r\n<ol id=\"fs-id1170572512650\">\r\n \t<li>If [latex]f(x)[\/latex] increases without bound from both sides as [latex]x[\/latex] approaches [latex]a[\/latex], the limit is [latex]+\u221e[\/latex].\r\n<div id=\"fs-id1170572337784\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty[\/latex].<\/div><\/li>\r\n \t<li>If [latex]f(x)[\/latex] decreases without bound from both sides as [latex]x[\/latex] approaches [latex]a[\/latex], the limit is [latex]-\u221e[\/latex].\r\n<div id=\"fs-id1170572337871\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=\u2212\\infty[\/latex].<\/div><\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox proTip\">It is important to understand that when we write statements such as [latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty [\/latex] or [latex]\\underset{x\\to a}{\\lim}f(x)=\u2212\\infty [\/latex] we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists.\r\n\r\nFor the limit of a function [latex]f(x)[\/latex] to exist at [latex]a[\/latex], it must approach a real number [latex]L[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex]. That said, if, for example, [latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty[\/latex], we always write [latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty [\/latex] rather than [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] DNE.\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]218963[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170571611160\">Evaluate each of the following limits, if possible. Use a table of functional values and graph [latex]f(x)=\\frac{1}{x}[\/latex] to confirm your conclusion.<\/p>\r\n\r\n<ol id=\"fs-id1170571611187\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572346978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572346978\"]\r\n<p id=\"fs-id1170572346978\">Begin by constructing a table of functional values.<\/p>\r\n\r\n<table id=\"fs-id1170572346981\" summary=\"Two tables side by side, each with two columns and seven rows. The headers are the same, x and 1\/x in the first row. In the first table, the values in the first column under x are -.01, -0.01, -0.001, -0.0001, -0.00001, and -0.000001. The values in the second column under the header are -10, -100, -1000, -10,000, -100,000, and -1,000,000. In the second column, the values in the first column under x are 0.1, 0.01, 0.001, 0.0001, 0.00001 and 0.000001. The values in the second column under the header are 10, 100, 1000, 10,000, 100,000, 1,000,000.\"><caption>Table of Functional Values for [latex]f(x)=\\frac{1}{x}[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{1}{x}[\/latex]<\/th>\r\n<th><\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{1}{x}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.1[\/latex]<\/td>\r\n<td>[latex]\u221210[\/latex]<\/td>\r\n<td rowspan=\"6\"><\/td>\r\n<td>[latex]0.1[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.01[\/latex]<\/td>\r\n<td>[latex]\u2212100[\/latex]<\/td>\r\n<td>[latex]0.01[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.001[\/latex]<\/td>\r\n<td>[latex]\u22121000[\/latex]<\/td>\r\n<td>[latex]0.001[\/latex]<\/td>\r\n<td>[latex]1000[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.0001[\/latex]<\/td>\r\n<td>[latex]\u221210,000[\/latex]<\/td>\r\n<td>[latex]0.0001[\/latex]<\/td>\r\n<td>[latex]10,000[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.00001[\/latex]<\/td>\r\n<td>[latex]\u2212100,000[\/latex]<\/td>\r\n<td>[latex]0.00001[\/latex]<\/td>\r\n<td>[latex]100,000[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.000001[\/latex]<\/td>\r\n<td>[latex]\u22121,000,000[\/latex]<\/td>\r\n<td>[latex]0.000001[\/latex]<\/td>\r\n<td>[latex]1,000,000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol id=\"fs-id1170571573960\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>The values of [latex]\\frac{1}{x}[\/latex] decrease without bound as [latex]x[\/latex] approaches [latex]0[\/latex] from the left. We conclude that\r\n<div id=\"fs-id1170572560361\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x}=\u2212\\infty[\/latex].<\/div><\/li>\r\n \t<li>The values of [latex]\\frac{1}{x}[\/latex] increase without bound as [latex]x[\/latex] approaches [latex]0[\/latex] from the right. We conclude that\r\n<div id=\"fs-id1170572560419\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x}=+\\infty[\/latex].<\/div><\/li>\r\n \t<li>Since [latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x}=\u2212\\infty [\/latex] and [latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x}=+\\infty [\/latex] have different values, we conclude that\r\n<div id=\"fs-id1170571596216\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x}[\/latex] DNE.<\/div><\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170571596248\">The graph of [latex]f(x)=\\frac{1}{x}[\/latex] in Figure 8 confirms these conclusions.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202912\/CNX_Calc_Figure_02_02_012.jpg\" alt=\"The graph of the function f(x) = 1\/x. The function curves asymptotically towards x=0 and y=0 in quadrants one and three.\" width=\"325\" height=\"427\" \/> Figure 8. The graph of [latex]f(x)=\\frac{1}{x}[\/latex] confirms that the limit as [latex]x[\/latex] approaches 0 does not exist.[\/caption][\/hidden-answer]<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Evaluate limits at infinity<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Infinite Limits<\/h2>\n<p>Evaluating limits, whether at a specific point or as we approach it from a particular direction, helps us understand how functions behave near that point. While some functions have limits that are finite numbers, others grow without bound\u2014these are cases of infinite limits.<\/p>\n<div id=\"fs-id1170571611973\" class=\"bc-section section\">\n<p>We now turn our attention to [latex]h(x)=\\frac{1}{(x-2)^2}[\/latex] (Figure 1 part c).<\/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 [latex]x[\/latex] gets closer to [latex]2[\/latex], [latex]h(x)[\/latex] increases without limit. This unbounded growth means that as [latex]x[\/latex] approaches [latex]2[\/latex], [latex]h(x)[\/latex] heads towards positive infinity, which we denote as:<\/p>\n<div id=\"fs-id1170571612232\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\lim}h(x)=+\\infty[\/latex]<\/div>\n<p id=\"fs-id1170571612271\">Infinite limits can be understood through the following general definitions.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>infinite limits<\/h3>\n<p id=\"fs-id1170571612290\"><strong>Infinite limits from the left:<\/strong> For a function [latex]f(x)[\/latex] within an interval that ends at [latex]a[\/latex], we say:<\/p>\n<ol id=\"fs-id1170571562562\">\n<li>The limit is [latex]+\u221e[\/latex] if [latex]f(x)[\/latex] increases without bound as [latex]x[\/latex] approaches [latex]a[\/latex] from the left.\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}f(x)=+\\infty[\/latex].<\/div>\n<\/li>\n<li>The limit is [latex]-\u221e[\/latex] if [latex]f(x)[\/latex] decreases without bound as [latex]x[\/latex] approaches [latex]a[\/latex] from the left.\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}f(x)=\u2212\\infty[\/latex].<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1170572346754\"><strong>Infinite limits from the right:<\/strong> For a function [latex]f(x)[\/latex] within an interval that ends at [latex]a[\/latex], we say:<\/p>\n<ol id=\"fs-id1170572346792\">\n<li>The limit is [latex]+\u221e[\/latex] if [latex]f(x)[\/latex] increases without bound as [latex]x[\/latex] approaches [latex]a[\/latex] from the right.\n<div id=\"fs-id1170572559800\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=+\\infty[\/latex].<\/div>\n<\/li>\n<li>The limit is [latex]-\u221e[\/latex] if [latex]f(x)[\/latex] decreases without bound as [latex]x[\/latex] approaches [latex]a[\/latex] from the right.\n<div id=\"fs-id1170572512575\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=\u2212\\infty[\/latex].<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1170572512615\"><strong>Two-sided infinite limit: <\/strong>For a function [latex]f(x)[\/latex] defined at all points except at [latex]a[\/latex]:<\/p>\n<ol id=\"fs-id1170572512650\">\n<li>If [latex]f(x)[\/latex] increases without bound from both sides as [latex]x[\/latex] approaches [latex]a[\/latex], the limit is [latex]+\u221e[\/latex].\n<div id=\"fs-id1170572337784\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty[\/latex].<\/div>\n<\/li>\n<li>If [latex]f(x)[\/latex] decreases without bound from both sides as [latex]x[\/latex] approaches [latex]a[\/latex], the limit is [latex]-\u221e[\/latex].\n<div id=\"fs-id1170572337871\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=\u2212\\infty[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox proTip\">It is important to understand that when we write statements such as [latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty[\/latex] or [latex]\\underset{x\\to a}{\\lim}f(x)=\u2212\\infty[\/latex] we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists.<\/p>\n<p>For the limit of a function [latex]f(x)[\/latex] to exist at [latex]a[\/latex], it must approach a real number [latex]L[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex]. That said, if, for example, [latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty[\/latex], we always write [latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty[\/latex] rather than [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] DNE.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm218963\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218963&theme=lumen&iframe_resize_id=ohm218963&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571611160\">Evaluate each of the following limits, if possible. Use a table of functional values and graph [latex]f(x)=\\frac{1}{x}[\/latex] to confirm your conclusion.<\/p>\n<ol id=\"fs-id1170571611187\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572346978\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572346978\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572346978\">Begin by constructing a table of functional values.<\/p>\n<table id=\"fs-id1170572346981\" summary=\"Two tables side by side, each with two columns and seven rows. The headers are the same, x and 1\/x in the first row. In the first table, the values in the first column under x are -.01, -0.01, -0.001, -0.0001, -0.00001, and -0.000001. The values in the second column under the header are -10, -100, -1000, -10,000, -100,000, and -1,000,000. In the second column, the values in the first column under x are 0.1, 0.01, 0.001, 0.0001, 0.00001 and 0.000001. The values in the second column under the header are 10, 100, 1000, 10,000, 100,000, 1,000,000.\">\n<caption>Table of Functional Values for [latex]f(x)=\\frac{1}{x}[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{1}{x}[\/latex]<\/th>\n<th><\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{1}{x}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\u22120.1[\/latex]<\/td>\n<td>[latex]\u221210[\/latex]<\/td>\n<td rowspan=\"6\"><\/td>\n<td>[latex]0.1[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.01[\/latex]<\/td>\n<td>[latex]\u2212100[\/latex]<\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.001[\/latex]<\/td>\n<td>[latex]\u22121000[\/latex]<\/td>\n<td>[latex]0.001[\/latex]<\/td>\n<td>[latex]1000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.0001[\/latex]<\/td>\n<td>[latex]\u221210,000[\/latex]<\/td>\n<td>[latex]0.0001[\/latex]<\/td>\n<td>[latex]10,000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.00001[\/latex]<\/td>\n<td>[latex]\u2212100,000[\/latex]<\/td>\n<td>[latex]0.00001[\/latex]<\/td>\n<td>[latex]100,000[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.000001[\/latex]<\/td>\n<td>[latex]\u22121,000,000[\/latex]<\/td>\n<td>[latex]0.000001[\/latex]<\/td>\n<td>[latex]1,000,000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol id=\"fs-id1170571573960\" style=\"list-style-type: lower-alpha;\">\n<li>The values of [latex]\\frac{1}{x}[\/latex] decrease without bound as [latex]x[\/latex] approaches [latex]0[\/latex] from the left. We conclude that\n<div id=\"fs-id1170572560361\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x}=\u2212\\infty[\/latex].<\/div>\n<\/li>\n<li>The values of [latex]\\frac{1}{x}[\/latex] increase without bound as [latex]x[\/latex] approaches [latex]0[\/latex] from the right. We conclude that\n<div id=\"fs-id1170572560419\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x}=+\\infty[\/latex].<\/div>\n<\/li>\n<li>Since [latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x}=\u2212\\infty[\/latex] and [latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x}=+\\infty[\/latex] have different values, we conclude that\n<div id=\"fs-id1170571596216\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x}[\/latex] DNE.<\/div>\n<\/li>\n<\/ol>\n<p id=\"fs-id1170571596248\">The graph of [latex]f(x)=\\frac{1}{x}[\/latex] in Figure 8 confirms these conclusions.<\/p>\n<figure style=\"width: 325px\" 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\/11202912\/CNX_Calc_Figure_02_02_012.jpg\" alt=\"The graph of the function f(x) = 1\/x. The function curves asymptotically towards x=0 and y=0 in quadrants one and three.\" width=\"325\" height=\"427\" \/><figcaption class=\"wp-caption-text\">Figure 8. 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