{"id":194,"date":"2023-09-20T22:48:11","date_gmt":"2023-09-20T22:48:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/infinite-limits\/"},"modified":"2025-04-03T18:09:26","modified_gmt":"2025-04-03T18:09:26","slug":"introduction-to-the-limit-of-a-function-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-the-limit-of-a-function-learn-it-4\/","title":{"raw":"Introduction to the Limit of a Function: Learn It 4","rendered":"Introduction to the Limit of a Function: Learn It 4"},"content":{"raw":"<h2>Infinite Limits<\/h2>\r\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>\r\n<div id=\"fs-id1170571611973\" class=\"bc-section section\">\r\n<p>We now turn our attention to [latex]h(x)=\\frac{1}{(x-2)^2}[\/latex] (Figure 1 part c).<\/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]\r\n\r\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>\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<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<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.<br \/>\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.<br \/>\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<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.<br \/>\r\n<div id=\"fs-id1170572559800\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=+\\infty[\/latex].<\/div>\r\n<\/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.<br \/>\r\n<div id=\"fs-id1170572512575\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a^+}{\\lim}f(x)=\u2212\\infty[\/latex].<\/div>\r\n<\/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<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].<br \/>\r\n<div id=\"fs-id1170572337784\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty[\/latex].<\/div>\r\n<\/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].<br \/>\r\n<div id=\"fs-id1170572337871\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=\u2212\\infty[\/latex].<\/div>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>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>\r\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>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]218963[\/ohm_question]<\/p>\r\n<\/section>\r\n<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<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<p>[reveal-answer q=\"fs-id1170572346978\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572346978\"]<\/p>\r\n<p id=\"fs-id1170572346978\">Begin by constructing a table of functional values.<\/p>\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.\">\r\n<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>\u00a0<\/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\">\u00a0<\/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<br \/>\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>\r\n<\/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<br \/>\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>\r\n<\/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<br \/>\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>\r\n<\/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]\r\n[\/hidden-answer]<\/section>\r\n<h3>Vertical Asymptotes<\/h3>\r\n<p id=\"fs-id1170572611930\">Continuing our exploration of infinite limits, it is worth highlighting functions like [latex]f(x)=\\dfrac{1}{(x-a)^n}[\/latex], where [latex]n[\/latex] is a positive integer, to illustrate the concept further. Such functions exhibit notable behavior as [latex]x[\/latex] approaches [latex]a[\/latex]: they tend toward infinity. Whether approaching from the left or the right, these functions have no finite limit at [latex]a[\/latex], and instead, they soar towards positive or negative infinity, depending on the parity of [latex]n[\/latex]. This tendency is depicted in Figure 9.<\/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\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>asymptotic behavior of power functions<\/h3>\r\n<p id=\"fs-id1170571654222\">If [latex]n[\/latex] is a positive even integer, then<\/p>\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<p>&nbsp;<\/p>\r\n<p id=\"fs-id1170571654279\">If [latex]n[\/latex] is a positive odd integer, then<\/p>\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<p>&nbsp;<\/p>\r\n<p id=\"fs-id1170571654339\">and<\/p>\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<\/section>\r\n<p id=\"fs-id1170571603730\">As we see in the graph, as [latex]x[\/latex] approaches the value of [latex]a[\/latex], regardless of the direction, the function values either increase or decrease sharply. This rapid change is visually represented by how the function's curve approaches, but never touches, the vertical line [latex]x=a[\/latex]. This specific line is known as a <strong>vertical asymptote<\/strong>, which is a boundary the function will never cross or reach.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>vertical asymptote<\/h3>\r\n<p>A vertical asymptote is a line that a function approaches but never intersects or reaches as the inputs get infinitely close to a particular point. <br \/>\r\n<br \/>\r\n<\/p>\r\n<p style=\"margin-left: 25px;\">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<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<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571656624\">Evaluate each of the following limits. Identify any vertical asymptotes of the function [latex]f(x)=\\dfrac{1}{(x+3)^4}[\/latex].<\/p>\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<p>[reveal-answer q=\"fs-id1170572632998\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572632998\"]<\/p>\r\n<p id=\"fs-id1170572632998\">We can use the limits summarized under Figure 9 directly.<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\">\r\n<p>[caption]Watch the following video to see the more examples of finding a vertical asymptote.[\/caption]<\/p>\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\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]<br \/>\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.<\/p>\r\n<p>You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction841to944_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).[\/hidden-answer]<\/p>\r\n<\/section>\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<section class=\"textbox example\">\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<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]\r\n[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]\r\n<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1170571610257\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571610257\"]<\/p>\r\n<p id=\"fs-id1170571610257\">Using the example above and the graph for reference, we arrive at the following values:<\/p>\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\r\nWatch the following video to see the worked solution to this example.<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\" data-mce-fragment=\"1\"><\/iframe><\/center>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\n<p>You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction1008to1197_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]20441[\/ohm_question]<\/p>\r\n<\/section>\r\n<\/div>","rendered":"<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\">\n<p>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\">\n<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><br \/>\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<p><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>\u00a0<\/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\">\u00a0<\/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. The graph of [latex]f(x)=\\frac{1}{x}[\/latex] confirms that the limit as [latex]x[\/latex] approaches 0 does not exist.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<h3>Vertical Asymptotes<\/h3>\n<p id=\"fs-id1170572611930\">Continuing our exploration of infinite limits, it is worth highlighting functions like [latex]f(x)=\\dfrac{1}{(x-a)^n}[\/latex], where [latex]n[\/latex] is a positive integer, to illustrate the concept further. Such functions exhibit notable behavior as [latex]x[\/latex] approaches [latex]a[\/latex]: they tend toward infinity. Whether approaching from the left or the right, these functions have no finite limit at [latex]a[\/latex], and instead, they soar towards positive or negative infinity, depending on the parity of [latex]n[\/latex]. This tendency is depicted in Figure 9.<\/p>\n<figure style=\"width: 731px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 9. The function [latex]f(x)=1\/(x-a)^n[\/latex] has infinite limits at [latex]a[\/latex].<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\">\n<h3>asymptotic behavior of power functions<\/h3>\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<\/section>\n<p id=\"fs-id1170571603730\">As we see in the graph, as [latex]x[\/latex] approaches the value of [latex]a[\/latex], regardless of the direction, the function values either increase or decrease sharply. This rapid change is visually represented by how the function&#8217;s curve approaches, but never touches, the vertical line [latex]x=a[\/latex]. This specific line is known as a <strong>vertical asymptote<\/strong>, which is a boundary the function will never cross or reach.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>vertical asymptote<\/h3>\n<p>A vertical asymptote is a line that a function approaches but never intersects or reaches as the inputs get infinitely close to a particular point. <\/p>\n<p style=\"margin-left: 25px;\">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<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571656624\">Evaluate each of the following limits. 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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572632998\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox watchIt\">\nWatch 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\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/button><\/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 transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction841to944_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).<\/div>\n<\/div>\n<\/section>\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<section class=\"textbox example\">\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]<br \/>\n<figure style=\"width: 342px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 10. The graph shows [latex]f(x)[\/latex].<\/figcaption><\/figure>\n<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571610257\">Show Solution<\/button><\/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<p>Watch the following video to see the worked solution to this example.<\/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\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>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 transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction1008to1197_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm20441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20441&theme=lumen&iframe_resize_id=ohm20441&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<\/div>\n","protected":false},"author":6,"menu_order":14,"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":484,"module-header":"learn_it","content_attributions":[{"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":""}],"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/194"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":24,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/194\/revisions"}],"predecessor-version":[{"id":4690,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/194\/revisions\/4690"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/484"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/194\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=194"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=194"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=194"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}