{"id":282,"date":"2021-02-04T00:47:44","date_gmt":"2021-02-04T00:47:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=282"},"modified":"2022-03-11T21:52:09","modified_gmt":"2022-03-11T21:52:09","slug":"infinite-limits","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/infinite-limits\/","title":{"raw":"Infinite Limits","rendered":"Infinite Limits"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Using correct notation, describe an infinite limit&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6145,&quot;3&quot;:{&quot;1&quot;:0},&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Using correct notation, describe an infinite limit<\/span><\/li>\r\n \t<li>Define a vertical asymptote<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2><span style=\"font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial; color: #373d3f;\">Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.<\/span><\/h2>\r\n<div id=\"fs-id1170571611973\" class=\"bc-section section\">\r\n<p id=\"fs-id1170571611984\">We now turn our attention to [latex]h(x)=\\frac{1}{(x-2)^2}[\/latex]. From its graph we see that as the values of [latex]x[\/latex] approach 2, the values of [latex]h(x)=\\frac{1}{(x-2)^2}[\/latex] become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of [latex]h(x)[\/latex] as [latex]x[\/latex] approaches 2 is positive infinity. Symbolically, we express this idea as<\/p>\r\n\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&nbsp;\r\n<p id=\"fs-id1170571612271\">More generally, we define infinite limits as follows:<\/p>\r\n\r\n<div id=\"fs-id1170571612277\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170571612282\">We define three types of <strong>infinite limits<\/strong>.<\/p>\r\n<p id=\"fs-id1170571612290\"><strong>Infinite limits from the left:<\/strong> Let [latex]f(x)[\/latex] be a function defined at all values in an open interval of the form [latex](b,a)[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170571562562\">\r\n \t<li>If the values of [latex]f(x)[\/latex] increase without bound as the values of [latex]x[\/latex] (where [latex]x&lt;a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the left is positive infinity and we write\r\n<div id=\"fs-id1170571562619\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}f(x)=+\\infty[\/latex].<\/div><\/li>\r\n \t<li>If the values of [latex]f(x)[\/latex] decrease without bound as the values of [latex]x[\/latex] (where [latex]x&lt;a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the left is negative infinity and we write\r\n<div id=\"fs-id1170572346714\" 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-id1170572346754\"><strong>Infinite limits 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].<\/p>\r\n\r\n<ol id=\"fs-id1170572346792\">\r\n \t<li>If the values of [latex]f(x)[\/latex] increase without bound as the values of [latex]x[\/latex] (where [latex]x&gt;a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the right is positive infinity and we write\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>If the values of [latex]f(x)[\/latex] decrease without bound as the values of [latex]x[\/latex] (where [latex]x&gt;a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the right is negative infinity and we write\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>Let [latex]f(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] in an open interval containing [latex]a[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572512650\">\r\n \t<li>If the values of [latex]f(x)[\/latex] increase without bound as the values of [latex]x[\/latex] (where [latex]x\\ne a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] is positive infinity and we write\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 the values of [latex]f(x)[\/latex] decrease without bound as the values of [latex]x[\/latex] (where [latex]x\\ne a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] is negative infinity and we write\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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]218963[\/ohm_question]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572337910\">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. 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>\r\n\r\n<div id=\"fs-id1170571611150\" class=\"textbook exercises\">\r\n<h3>Example: Recognizing an <strong>Infinite Limit<\/strong><\/h3>\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)=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>\u22120.1<\/td>\r\n<td>\u221210<\/td>\r\n<td rowspan=\"6\"><\/td>\r\n<td>0.1<\/td>\r\n<td>10<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.01<\/td>\r\n<td>\u2212100<\/td>\r\n<td>0.01<\/td>\r\n<td>100<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.001<\/td>\r\n<td>\u22121000<\/td>\r\n<td>0.001<\/td>\r\n<td>1000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.0001<\/td>\r\n<td>\u221210,000<\/td>\r\n<td>0.0001<\/td>\r\n<td>10,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.00001<\/td>\r\n<td>\u2212100,000<\/td>\r\n<td>0.00001<\/td>\r\n<td>100,000<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.000001<\/td>\r\n<td>\u22121,000,000<\/td>\r\n<td>0.000001<\/td>\r\n<td>1,000,000<\/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 0 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 0 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]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571596330\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571596338\">Evaluate each of the following limits, if possible. Use a table of functional values and graph [latex]f(x)=\\dfrac{1}{x^2}[\/latex] to confirm your conclusion.<\/p>\r\n\r\n<ol id=\"fs-id1170571612847\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"273990\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"273990\"]\r\n<p id=\"fs-id1170571612943\">Follow the procedures from the example above.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170571612954\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571612954\"]\r\n<p id=\"fs-id1170571612954\">a. [latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex];<\/p>\r\nb. [latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex];\r\n\r\nc. [latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x^2}=+\\infty [\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572611930\">It is useful to point out that functions of the form [latex]f(x)=\\dfrac{1}{(x-a)^n}[\/latex], where [latex]n[\/latex] is a positive integer, have infinite limits as [latex]x[\/latex] approaches [latex]a[\/latex] from either the left or right (Figure 9). These limits are summarized below the graphs.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202915\/CNX_Calc_Figure_02_02_014.jpg\" alt=\"Two graphs side by side of f(x) = 1 \/ (x-a)^n. The first graph shows the case where n is an odd positive integer, and the second shows the case where n is an even positive integer. In the first, the graph has two segments. Each curve asymptotically towards the x axis, also known as y=0, and x=a. The segment to the left of x=a is below the x axis, and the segment to the right of x=a is above the x axis. In the second graph, both segments are above the x axis.\" width=\"731\" height=\"427\" \/> Figure 9. The function [latex]f(x)=1\/(x-a)^n[\/latex] has infinite limits at [latex]a[\/latex].[\/caption]\r\n<div id=\"fs-id1170571654206\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Infinite Limits from Positive Integers<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170571654222\">If [latex]n[\/latex] is a positive even integer, then<\/p>\r\n\r\n<div id=\"fs-id1170571654230\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}\\dfrac{1}{(x-a)^n}=+\\infty[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170571654279\">If [latex]n[\/latex] is a positive odd integer, then<\/p>\r\n\r\n<div id=\"fs-id1170571654287\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}\\dfrac{1}{(x-a)^n}=+\\infty [\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170571654339\">and<\/p>\r\n\r\n<div id=\"fs-id1170571654342\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}\\dfrac{1}{(x-a)^n}=\u2212\\infty[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170571603730\">We should also point out that in the graphs of [latex]f(x)=\\dfrac{1}{(x-a)^n}[\/latex], points on the graph having [latex]x[\/latex]-coordinates very near to [latex]a[\/latex] are very close to the vertical line [latex]x=a[\/latex]. That is, as [latex]x[\/latex] approaches [latex]a[\/latex], the points on the graph of [latex]f(x)[\/latex] are closer to the line [latex]x=a[\/latex]. The line [latex]x=a[\/latex] is called a <strong>vertical asymptote<\/strong> of the graph. We formally define a vertical asymptote as follows:<\/p>\r\n\r\n<div id=\"fs-id1170571603845\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n<p id=\"fs-id1170571656454\">Let [latex]f(x)[\/latex] be a function. If any of the following conditions hold, then the line [latex]x=a[\/latex] is a <strong>vertical asymptote<\/strong> of [latex]f(x)[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1165042770942\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\underset{x\\to a^-}{\\lim}f(x)&amp; =\\hfill &amp; +\\infty \\, \\text{or} \\, -\\infty \\hfill \\\\ \\hfill \\underset{x\\to a^+}{\\lim}f(x)&amp; =\\hfill &amp; +\\infty \\, \\text{or} \\, \u2212\\infty \\hfill \\\\ &amp; \\text{or}\\hfill &amp; \\\\ \\hfill \\underset{x\\to a}{\\lim}f(x)&amp; =\\hfill &amp; +\\infty \\, \\text{or} \\, \u2212\\infty \\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571656614\" class=\"textbook exercises\">\r\n<h3>Example: Finding a Vertical Asymptote<\/h3>\r\n<p id=\"fs-id1170571656624\">Evaluate each of the following limits using the limits summarized under Figure 9. Identify any vertical asymptotes of the function [latex]f(x)=\\dfrac{1}{(x+3)^4}[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572388087\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to -3^-}{\\lim}\\dfrac{1}{(x+3)^4}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to -3^+}{\\lim}\\dfrac{1}{(x+3)^4}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to -3}{\\lim}\\dfrac{1}{(x+3)^4}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572632998\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572632998\"]\r\n<p id=\"fs-id1170572632998\">We can use the limits summarized under Figure 9 directly.<\/p>\r\n\r\n<ol id=\"fs-id1170572633005\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to -3^-}{\\lim}\\frac{1}{(x+3)^4}=+\\infty [\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to -3^+}{\\lim}\\frac{1}{(x+3)^4}=+\\infty [\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to -3}{\\lim}\\frac{1}{(x+3)^4}=+\\infty [\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170571652076\">The function [latex]f(x)=\\frac{1}{(x+3)^4}[\/latex] has a vertical asymptote of [latex]x=-3[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the more examples of finding a vertical asymptote.[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qiHi41CfnFA?controls=0&amp;start=841&amp;end=944&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\/2.2TheLimitOfAFunction841to944_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-id1170571652132\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571652140\">Evaluate each of the following limits. Identify any vertical asymptotes of the function [latex]f(x)=\\dfrac{1}{(x-2)^3}[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170571652179\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 2^+}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 2}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"8356701\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"8356701\"]\r\n<p id=\"fs-id1170571545540\">Use the limits summarized under Figure 9.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170571545551\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571545551\"]\r\n<p id=\"fs-id1170571545551\">a. [latex]\\underset{x\\to 2^-}{\\lim}\\frac{1}{(x-2)^3}=\u2212\\infty[\/latex];<\/p>\r\nb. [latex]\\underset{x\\to 2^+}{\\lim}\\frac{1}{(x-2)^3}=+\\infty[\/latex];\r\n\r\nc. [latex]\\underset{x\\to 2}{\\lim}\\frac{1}{(x-2)^3}[\/latex] DNE. The line [latex]x=2[\/latex] is the vertical asymptote of [latex]f(x)=\\frac{1}{(x-2)^3}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572642379\">In the next example, we put our knowledge of various types of limits to use to analyze the behavior of a function at several different points.<\/p>\r\n\r\n<div id=\"fs-id1170572642384\" class=\"textbook exercises\">\r\n<h3>Example: Behavior of a Function at Different Points<\/h3>\r\n<p id=\"fs-id1170572642393\">Use the graph of [latex]f(x)[\/latex] in Figure 10 to determine each of the following values:<\/p>\r\n\r\n<ol id=\"fs-id1170572642414\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to -4^-}{\\lim}f(x); \\, \\underset{x\\to -4^+}{\\lim}f(x); \\, \\underset{x\\to -4}{\\lim}f(x); \\, f(-4)[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to -2^-}{\\lim}f(x); \\, \\underset{x\\to -2^+}{\\lim}f(x); \\, \\underset{x\\to -2}{\\lim}f(x); \\, f(-2)[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 1^-}{\\lim}f(x); \\, \\underset{x\\to 1^+}{\\lim}f(x); \\, \\underset{x\\to 1}{\\lim}f(x); \\, f(1)[\/latex][caption id=\"\" align=\"aligncenter\" width=\"342\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202918\/CNX_Calc_Figure_02_02_015.jpg\" alt=\"The graph of a function f(x) described by the above limits and values. There is a smooth curve for values below x=-2; at (-2, 3), there is an open circle. There is a smooth curve between (-2, 1] with a closed circle at (1,6). There is an open circle at (1,3), and a smooth curve stretching from there down asymptotically to negative infinity along x=3. The function also curves asymptotically along x=3 on the other side, also stretching to negative infinity. The function then changes concavity in the first quadrant around y=4.5 and continues up.\" width=\"342\" height=\"347\" \/> Figure 10. The graph shows [latex]f(x)[\/latex].[\/caption]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170571610257\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571610257\"]\r\n<p id=\"fs-id1170571610257\">Using the example above and the graph for reference, we arrive at the following values:<\/p>\r\n\r\n<ol id=\"fs-id1170571610264\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to -4^-}{\\lim}f(x)=0; \\, \\underset{x\\to -4^+}{\\lim}f(x)=0; \\, \\underset{x\\to -4}{\\lim}f(x)=0; \\, f(-4)=0[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to -2^-}{\\lim}f(x)=3; \\, \\underset{x\\to -2^+}{\\lim}f(x)=3; \\, \\underset{x\\to -2}{\\lim}f(x)=3; \\, f(-2)[\/latex] is undefined<\/li>\r\n \t<li>[latex]\\underset{x\\to 1^-}{\\lim}f(x)=6; \\, \\underset{x\\to 1^+}{\\lim}f(x)=3; \\, \\underset{x\\to 1}{\\lim}f(x)=DNE[\/latex]; [latex]f(1)=6[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Behavior of a Function at Different Points. Note this video also repeats the steps for x = 3.[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qiHi41CfnFA?controls=0&amp;start=1008&amp;end=1197&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]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\/2.2TheLimitOfAFunction1008to1197_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-id1170572624466\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572624475\">Evaluate [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex] for [latex]f(x)[\/latex] shown here:<\/p>\r\n\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\/11202920\/CNX_Calc_Figure_02_02_016.jpg\" alt=\"A graph of a piecewise function. The first segment curves from the third quadrant to the first, crossing through the second quadrant. Where the endpoint would be in the first quadrant is an open circle. The second segment starts at a closed circle a few units below the open circle. It curves down from quadrant one to quadrant four.\" width=\"487\" height=\"350\" \/> Figure 11.[\/caption]\r\n\r\n[reveal-answer q=\"834551\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"834551\"]\r\n<p id=\"fs-id1170572624532\">Compare the limit from the right with the limit from the left.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572624538\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572624538\"]\r\n<p id=\"fs-id1170572624538\">Does not exist (DNE).<\/p>\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]20441[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Using correct notation, describe an infinite limit&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6145,&quot;3&quot;:{&quot;1&quot;:0},&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Using correct notation, describe an infinite limit<\/span><\/li>\n<li>Define a vertical asymptote<\/li>\n<\/ul>\n<\/div>\n<h2><span style=\"font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial; color: #373d3f;\">Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.<\/span><\/h2>\n<div id=\"fs-id1170571611973\" class=\"bc-section section\">\n<p id=\"fs-id1170571611984\">We now turn our attention to [latex]h(x)=\\frac{1}{(x-2)^2}[\/latex]. From its graph we see that as the values of [latex]x[\/latex] approach 2, the values of [latex]h(x)=\\frac{1}{(x-2)^2}[\/latex] become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of [latex]h(x)[\/latex] as [latex]x[\/latex] approaches 2 is positive infinity. Symbolically, we express this idea 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>&nbsp;<\/p>\n<p id=\"fs-id1170571612271\">More generally, we define infinite limits as follows:<\/p>\n<div id=\"fs-id1170571612277\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1170571612282\">We define three types of <strong>infinite limits<\/strong>.<\/p>\n<p id=\"fs-id1170571612290\"><strong>Infinite limits from the left:<\/strong> Let [latex]f(x)[\/latex] be a function defined at all values in an open interval of the form [latex](b,a)[\/latex].<\/p>\n<ol id=\"fs-id1170571562562\">\n<li>If the values of [latex]f(x)[\/latex] increase without bound as the values of [latex]x[\/latex] (where [latex]x<a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the left is positive infinity and we write\n\n\n<div id=\"fs-id1170571562619\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}f(x)=+\\infty[\/latex].<\/div>\n<\/li>\n<li>If the values of [latex]f(x)[\/latex] decrease without bound as the values of [latex]x[\/latex] (where [latex]x<a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the left is negative infinity and we write\n\n\n<div id=\"fs-id1170572346714\" 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-id1170572346754\"><strong>Infinite limits 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].<\/p>\n<ol id=\"fs-id1170572346792\">\n<li>If the values of [latex]f(x)[\/latex] increase without bound as the values of [latex]x[\/latex] (where [latex]x>a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the right is positive infinity and we write\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>If the values of [latex]f(x)[\/latex] decrease without bound as the values of [latex]x[\/latex] (where [latex]x>a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the right is negative infinity and we write\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>Let [latex]f(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] in an open interval containing [latex]a[\/latex].<\/p>\n<ol id=\"fs-id1170572512650\">\n<li>If the values of [latex]f(x)[\/latex] increase without bound as the values of [latex]x[\/latex] (where [latex]x\\ne a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] is positive infinity and we write\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 the values of [latex]f(x)[\/latex] decrease without bound as the values of [latex]x[\/latex] (where [latex]x\\ne a[\/latex]) approach the number [latex]a[\/latex], then we say that the limit as [latex]x[\/latex] approaches [latex]a[\/latex] is negative infinity and we write\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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm218963\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=218963&theme=oea&iframe_resize_id=ohm218963&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p id=\"fs-id1170572337910\">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. 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<div id=\"fs-id1170571611150\" class=\"textbook exercises\">\n<h3>Example: Recognizing an <strong>Infinite Limit<\/strong><\/h3>\n<p id=\"fs-id1170571611160\">Evaluate each of the following limits, if possible. Use a table of functional values and graph [latex]f(x)=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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572346978\">Show Solution<\/span><\/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>\u22120.1<\/td>\n<td>\u221210<\/td>\n<td rowspan=\"6\"><\/td>\n<td>0.1<\/td>\n<td>10<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.01<\/td>\n<td>\u2212100<\/td>\n<td>0.01<\/td>\n<td>100<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.001<\/td>\n<td>\u22121000<\/td>\n<td>0.001<\/td>\n<td>1000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.0001<\/td>\n<td>\u221210,000<\/td>\n<td>0.0001<\/td>\n<td>10,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.00001<\/td>\n<td>\u2212100,000<\/td>\n<td>0.00001<\/td>\n<td>100,000<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.000001<\/td>\n<td>\u22121,000,000<\/td>\n<td>0.000001<\/td>\n<td>1,000,000<\/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 0 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 0 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<div style=\"width: 335px\" 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\" \/><\/p>\n<p class=\"wp-caption-text\">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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571596330\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571596338\">Evaluate each of the following limits, if possible. Use a table of functional values and graph [latex]f(x)=\\dfrac{1}{x^2}[\/latex] to confirm your conclusion.<\/p>\n<ol id=\"fs-id1170571612847\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q273990\">Hint<\/span><\/p>\n<div id=\"q273990\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571612943\">Follow the procedures from the example above.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571612954\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571612954\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571612954\">a. [latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex];<\/p>\n<p>b. [latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex];<\/p>\n<p>c. [latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572611930\">It is useful to point out that functions of the form [latex]f(x)=\\dfrac{1}{(x-a)^n}[\/latex], where [latex]n[\/latex] is a positive integer, have infinite limits as [latex]x[\/latex] approaches [latex]a[\/latex] from either the left or right (Figure 9). These limits are summarized below the graphs.<\/p>\n<div style=\"width: 741px\" 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\/11202915\/CNX_Calc_Figure_02_02_014.jpg\" alt=\"Two graphs side by side of f(x) = 1 \/ (x-a)^n. The first graph shows the case where n is an odd positive integer, and the second shows the case where n is an even positive integer. In the first, the graph has two segments. Each curve asymptotically towards the x axis, also known as y=0, and x=a. The segment to the left of x=a is below the x axis, and the segment to the right of x=a is above the x axis. In the second graph, both segments are above the x axis.\" width=\"731\" height=\"427\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9. The function [latex]f(x)=1\/(x-a)^n[\/latex] has infinite limits at [latex]a[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1170571654206\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Infinite Limits from Positive Integers<\/h3>\n<hr \/>\n<p id=\"fs-id1170571654222\">If [latex]n[\/latex] is a positive even integer, then<\/p>\n<div id=\"fs-id1170571654230\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}\\dfrac{1}{(x-a)^n}=+\\infty[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170571654279\">If [latex]n[\/latex] is a positive odd integer, then<\/p>\n<div id=\"fs-id1170571654287\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}\\dfrac{1}{(x-a)^n}=+\\infty[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170571654339\">and<\/p>\n<div id=\"fs-id1170571654342\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a^-}{\\lim}\\dfrac{1}{(x-a)^n}=\u2212\\infty[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1170571603730\">We should also point out that in the graphs of [latex]f(x)=\\dfrac{1}{(x-a)^n}[\/latex], points on the graph having [latex]x[\/latex]-coordinates very near to [latex]a[\/latex] are very close to the vertical line [latex]x=a[\/latex]. That is, as [latex]x[\/latex] approaches [latex]a[\/latex], the points on the graph of [latex]f(x)[\/latex] are closer to the line [latex]x=a[\/latex]. The line [latex]x=a[\/latex] is called a <strong>vertical asymptote<\/strong> of the graph. We formally define a vertical asymptote as follows:<\/p>\n<div id=\"fs-id1170571603845\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<p id=\"fs-id1170571656454\">Let [latex]f(x)[\/latex] be a function. If any of the following conditions hold, then the line [latex]x=a[\/latex] is a <strong>vertical asymptote<\/strong> of [latex]f(x)[\/latex]:<\/p>\n<div id=\"fs-id1165042770942\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\underset{x\\to a^-}{\\lim}f(x)& =\\hfill & +\\infty \\, \\text{or} \\, -\\infty \\hfill \\\\ \\hfill \\underset{x\\to a^+}{\\lim}f(x)& =\\hfill & +\\infty \\, \\text{or} \\, \u2212\\infty \\hfill \\\\ & \\text{or}\\hfill & \\\\ \\hfill \\underset{x\\to a}{\\lim}f(x)& =\\hfill & +\\infty \\, \\text{or} \\, \u2212\\infty \\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1170571656614\" class=\"textbook exercises\">\n<h3>Example: Finding a Vertical Asymptote<\/h3>\n<p id=\"fs-id1170571656624\">Evaluate each of the following limits using the limits summarized under Figure 9. Identify any vertical asymptotes of the function [latex]f(x)=\\dfrac{1}{(x+3)^4}[\/latex].<\/p>\n<ol id=\"fs-id1170572388087\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to -3^-}{\\lim}\\dfrac{1}{(x+3)^4}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to -3^+}{\\lim}\\dfrac{1}{(x+3)^4}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to -3}{\\lim}\\dfrac{1}{(x+3)^4}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572632998\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572632998\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572632998\">We can use the limits summarized under Figure 9 directly.<\/p>\n<ol id=\"fs-id1170572633005\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to -3^-}{\\lim}\\frac{1}{(x+3)^4}=+\\infty[\/latex]<\/li>\n<li>[latex]\\underset{x\\to -3^+}{\\lim}\\frac{1}{(x+3)^4}=+\\infty[\/latex]<\/li>\n<li>[latex]\\underset{x\\to -3}{\\lim}\\frac{1}{(x+3)^4}=+\\infty[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1170571652076\">The function [latex]f(x)=\\frac{1}{(x+3)^4}[\/latex] has a vertical asymptote of [latex]x=-3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the more examples of finding a vertical asymptote.<\/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=841&amp;end=944&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\/2.2TheLimitOfAFunction841to944_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-id1170571652132\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571652140\">Evaluate each of the following limits. Identify any vertical asymptotes of the function [latex]f(x)=\\dfrac{1}{(x-2)^3}[\/latex].<\/p>\n<ol id=\"fs-id1170571652179\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2^+}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q8356701\">Hint<\/span><\/p>\n<div id=\"q8356701\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571545540\">Use the limits summarized under Figure 9.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571545551\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571545551\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571545551\">a. [latex]\\underset{x\\to 2^-}{\\lim}\\frac{1}{(x-2)^3}=\u2212\\infty[\/latex];<\/p>\n<p>b. [latex]\\underset{x\\to 2^+}{\\lim}\\frac{1}{(x-2)^3}=+\\infty[\/latex];<\/p>\n<p>c. [latex]\\underset{x\\to 2}{\\lim}\\frac{1}{(x-2)^3}[\/latex] DNE. The line [latex]x=2[\/latex] is the vertical asymptote of [latex]f(x)=\\frac{1}{(x-2)^3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572642379\">In the next example, we put our knowledge of various types of limits to use to analyze the behavior of a function at several different points.<\/p>\n<div id=\"fs-id1170572642384\" class=\"textbook exercises\">\n<h3>Example: Behavior of a Function at Different Points<\/h3>\n<p id=\"fs-id1170572642393\">Use the graph of [latex]f(x)[\/latex] in Figure 10 to determine each of the following values:<\/p>\n<ol id=\"fs-id1170572642414\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to -4^-}{\\lim}f(x); \\, \\underset{x\\to -4^+}{\\lim}f(x); \\, \\underset{x\\to -4}{\\lim}f(x); \\, f(-4)[\/latex]<\/li>\n<li>[latex]\\underset{x\\to -2^-}{\\lim}f(x); \\, \\underset{x\\to -2^+}{\\lim}f(x); \\, \\underset{x\\to -2}{\\lim}f(x); \\, f(-2)[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 1^-}{\\lim}f(x); \\, \\underset{x\\to 1^+}{\\lim}f(x); \\, \\underset{x\\to 1}{\\lim}f(x); \\, f(1)[\/latex]\n<div style=\"width: 352px\" 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\/11202918\/CNX_Calc_Figure_02_02_015.jpg\" alt=\"The graph of a function f(x) described by the above limits and values. There is a smooth curve for values below x=-2; at (-2, 3), there is an open circle. There is a smooth curve between (-2, 1] with a closed circle at (1,6). There is an open circle at (1,3), and a smooth curve stretching from there down asymptotically to negative infinity along x=3. The function also curves asymptotically along x=3 on the other side, also stretching to negative infinity. The function then changes concavity in the first quadrant around y=4.5 and continues up.\" width=\"342\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 10. The graph shows [latex]f(x)[\/latex].<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571610257\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571610257\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571610257\">Using the example above and the graph for reference, we arrive at the following values:<\/p>\n<ol id=\"fs-id1170571610264\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to -4^-}{\\lim}f(x)=0; \\, \\underset{x\\to -4^+}{\\lim}f(x)=0; \\, \\underset{x\\to -4}{\\lim}f(x)=0; \\, f(-4)=0[\/latex]<\/li>\n<li>[latex]\\underset{x\\to -2^-}{\\lim}f(x)=3; \\, \\underset{x\\to -2^+}{\\lim}f(x)=3; \\, \\underset{x\\to -2}{\\lim}f(x)=3; \\, f(-2)[\/latex] is undefined<\/li>\n<li>[latex]\\underset{x\\to 1^-}{\\lim}f(x)=6; \\, \\underset{x\\to 1^+}{\\lim}f(x)=3; \\, \\underset{x\\to 1}{\\lim}f(x)=DNE[\/latex]; [latex]f(1)=6[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Behavior of a Function at Different Points. Note this video also repeats the steps for x = 3.<\/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=1008&amp;end=1197&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=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" 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\/2.2TheLimitOfAFunction1008to1197_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-id1170572624466\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572624475\">Evaluate [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex] for [latex]f(x)[\/latex] shown here:<\/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\/11202920\/CNX_Calc_Figure_02_02_016.jpg\" alt=\"A graph of a piecewise function. The first segment curves from the third quadrant to the first, crossing through the second quadrant. Where the endpoint would be in the first quadrant is an open circle. The second segment starts at a closed circle a few units below the open circle. It curves down from quadrant one to quadrant four.\" width=\"487\" height=\"350\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 11.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834551\">Hint<\/span><\/p>\n<div id=\"q834551\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572624532\">Compare the limit from the right with the limit from the left.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572624538\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572624538\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572624538\">Does not exist (DNE).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm20441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20441&theme=oea&iframe_resize_id=ohm20441&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-282\">\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":9,"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-282","chapter","type-chapter","status-publish","hentry"],"part":28,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/282","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":37,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/282\/revisions"}],"predecessor-version":[{"id":4770,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/282\/revisions\/4770"}],"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\/282\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=282"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=282"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=282"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=282"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}