{"id":279,"date":"2021-02-04T00:46:48","date_gmt":"2021-02-04T00:46:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=279"},"modified":"2022-03-11T21:51:09","modified_gmt":"2022-03-11T21:51:09","slug":"definition-of-a-limit","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/definition-of-a-limit\/","title":{"raw":"Definition of a Limit","rendered":"Definition of a Limit"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Using correct notation, describe the limit of a function<\/li>\r\n \t<li>Use a table of values to estimate the limit of a function or to identify when the limit does not exist<\/li>\r\n \t<li>Use a graph to estimate the limit of a function or to identify when the limit does not exist<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1170572450938\">We begin our exploration of limits by taking a look at the graphs of the functions<\/p>\r\n\r\n<div id=\"fs-id1170572346957\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)=\\dfrac{x^2-4}{x-2}, \\ \\, g(x)=\\dfrac{|x-2|}{x-2}[\/latex],\u00a0 and\u00a0 [latex]h(x)=\\dfrac{1}{(x-2)^2}[\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572216951\">which are shown in Figure 1. In particular, let\u2019s focus our attention on the behavior of each graph at and around [latex]x=2[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202849\/CNX_Calc_Figure_02_02_001.jpg\" alt=\"&quot;Three 2 and x= -1 for x &lt; 2. There are open circles at both endpoints (2, 1) and (-2, 1). The third is h(x) = 1 \/ (x-2)^2, in which the function curves asymptotically towards y=0 and x=2 in quadrants one and two.&quot; width=&quot;975&quot; height=&quot;434&quot;\" width=\"975\" height=\"434\" \/> Figure 1. These graphs show the behavior of three different functions around [latex]x=2[\/latex].[\/caption]\r\n<p id=\"fs-id1170572175064\">Each of the three functions is undefined at [latex]x=2[\/latex], but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of [latex]x=2[\/latex]. To express the behavior of each graph in the vicinity of 2 more completely, we need to introduce the concept of a limit.<\/p>\r\n\r\n<div id=\"fs-id1170572280146\" class=\"bc-section section\">\r\n<h2>Intuitive Definition of a Limit<\/h2>\r\n<p id=\"fs-id1170572449458\">Let\u2019s first take a closer look at how the function [latex]f(x)=\\dfrac{(x^2-4)}{(x-2)}[\/latex] behaves around [latex]x=2[\/latex] in Figure 1. As the values of [latex]x[\/latex] approach 2 from either side of 2, the values of [latex]y=f(x)[\/latex] approach 4. Mathematically, we say that the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches 2 is 4. Symbolically, we express this limit as<\/p>\r\n\r\n<div id=\"fs-id1170571655049\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x \\to 2}{\\lim}f(x)=4[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572220702\">From this very brief informal look at one limit, let\u2019s start to develop an <strong>intuitive definition of the limit<\/strong>. We can think of the limit of a function at a number [latex]a[\/latex] as being the one real number [latex]L[\/latex] that the functional values approach as the [latex]x[\/latex]-values approach [latex]a[\/latex]<em>,<\/em> provided such a real number [latex]L[\/latex] exists. Stated more carefully, we have the following definition:<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572151707\">Let [latex]f(x)[\/latex] be a function defined at all values in an open interval containing [latex]a[\/latex], with the possible exception of [latex]a[\/latex] itself, and let [latex]L[\/latex]\u00a0be a real number. If <em>all<\/em> values of the function [latex]f(x)[\/latex] approach the real number [latex]L[\/latex]\u00a0as the values of [latex]x(\\ne a)[\/latex] approach the number [latex]a[\/latex], then we say that the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is [latex]L[\/latex]. (More succinct, as [latex]x[\/latex] gets closer to [latex]a[\/latex], [latex]f(x)[\/latex] gets closer and stays close to [latex]L[\/latex].) Symbolically, we express this idea as<\/p>\r\n\r\n<div id=\"fs-id1170572133132\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]6241[\/ohm_question]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572244141\">We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy.<\/p>\r\n\r\n<div id=\"fs-id1170571656330\" class=\"textbox examples\">\r\n<h3>Problem-Solving Strategy: Evaluating a Limit Using a Table of Functional Values<\/h3>\r\n<ol id=\"fs-id1170572480841\">\r\n \t<li>To evaluate [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex], we begin by completing a table of functional values. We should choose two sets of [latex]x[\/latex]-values\u2014one set of values approaching [latex]a[\/latex] and less than [latex]a[\/latex], and another set of values approaching [latex]a[\/latex] and greater than [latex]a[\/latex]. The table below demonstrates what your tables might look like.\r\n<table id=\"fs-id1170572204940\" summary=\"There are two tables. They both have two columns and five rows. The first table has headers x and f(x) in the first row. Under x in the first column are the values a-0.1, a-0.01, a-0.001, and a-0.0001. Under f(x) in the second column are values f(a-0.1), f(a-0.01), f(a-0.001), and f(a-0.0001). At the bottom is a note that one may \u201cuse additional values as necessary\u201d in both columns. The second table has headers x and f(x) in the first row. Under x in the first column are the values a+0.1, a+0.01, a+0.001, and a+0.0001. Under f(x) in the second column are values f(a+0.1), f(a+0.01), f(a+0.001), and f(a+0.0001). At the bottom is a note that one may \u201cuse additional values as necessary\u201d in both columns.\"><caption>Table of Functional Values for [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<th><\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.1[\/latex]<\/td>\r\n<td>[latex]f(a-0.1)[\/latex]<\/td>\r\n<td rowspan=\"5\"><\/td>\r\n<td>[latex]a+0.1[\/latex]<\/td>\r\n<td>[latex]f(a+0.1)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.01[\/latex]<\/td>\r\n<td>[latex]f(a-0.01)[\/latex]<\/td>\r\n<td>[latex]a+0.01[\/latex]<\/td>\r\n<td>[latex]f(a+0.01)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.001[\/latex]<\/td>\r\n<td>[latex]f(a-0.001)[\/latex]<\/td>\r\n<td>[latex]a+0.001[\/latex]<\/td>\r\n<td>[latex]f(a+0.001)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.0001[\/latex]<\/td>\r\n<td>[latex]f(a-0.0001)[\/latex]<\/td>\r\n<td>[latex]a+0.0001[\/latex]<\/td>\r\n<td>[latex]f(a+0.0001)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td colspan=\"2\">Use additional values as necessary.<\/td>\r\n<td colspan=\"2\">Use additional values as necessary.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Next, let\u2019s look at the values in each of the [latex]f(x)[\/latex] columns and determine whether the values seem to be approaching a single value as we move down each column. In our columns, we look at the sequence [latex]f(a-0.1), \\, f(a-0.01), \\, f(a-0.001), \\, f(a-0.0001),[\/latex] and so on, and [latex]f(a+0.1), \\, f(a+0.01), \\, f(a+0.001), \\, f(a+0.0001)[\/latex] and so on. (<em>Note<\/em>: Although we have chosen the [latex]x[\/latex]-values [latex]a \\pm 0.1, \\, a \\pm 0.01, \\, a \\pm 0.001, \\, a \\pm 0.0001[\/latex], and so forth, and these values will probably work nearly every time, on very rare occasions we may need to modify our choices.)<\/li>\r\n \t<li>If both columns approach a common [latex]y[\/latex]-value [latex]L[\/latex], we state [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]. We can use the following strategy to confirm the result obtained from the table or as an alternative method for estimating a limit.<\/li>\r\n \t<li>Using a graphing calculator or computer software that allows us graph functions, we can plot the function [latex]f(x)[\/latex], making sure the functional values of [latex]f(x)[\/latex] for [latex]x[\/latex]-values near [latex]a[\/latex] are in our window. We can use the trace feature to move along the graph of the function and watch the [latex]y[\/latex]-value readout as the [latex]x[\/latex]-values approach [latex]a[\/latex]. If the [latex]y[\/latex]-values approach [latex]L[\/latex]\u00a0as our [latex]x[\/latex]-values approach [latex]a[\/latex] from both directions, then [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]. We may need to zoom in on our graph and repeat this process several times.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1170572175147\">We apply this Problem-Solving Strategy to compute a limit below.<\/p>\r\n\r\n<div id=\"fs-id1170572561451\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating a Limit Using a Table of Functional Values 1<\/h3>\r\n<p id=\"fs-id1170571596728\">Evaluate [latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin x}{x}[\/latex] using a table of functional values.<\/p>\r\n[reveal-answer q=\"fs-id1170572552454\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572552454\"]We have calculated the values of [latex]f(x)=\\dfrac{(\\sin x)}{x}[\/latex] for the values of [latex]x[\/latex] listed in the table below.\r\n<table id=\"fs-id1170572208852\" summary=\"There are two tables. They both have two columns and five rows. The first table has headers x and sin(x)\/x in the first row. Under x in the first column are the values -0.1, -0.01, -0.001, and -0.0001. Under sin(x)\/x in the second column are values 0.998334166468, 0.999983333417, 0.999999833333, and 0.999999998333. The second table has headers x and sin(x)\/x in the first row. Under x in the first column are the values 0.1, 0.01, 0.001, and 0.0001. Under sin(x)\/x in the second column are values 0.998334166468, 0.999983333417, 0.999999833333, and 0.999999998333.\"><caption>Table of Functional Values for [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{\\sin x}{x}[\/latex]<\/th>\r\n<th><\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{\\sin x}{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>0.998334166468<\/td>\r\n<td rowspan=\"4\"><\/td>\r\n<td>0.1<\/td>\r\n<td>0.998334166468<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.01<\/td>\r\n<td>0.999983333417<\/td>\r\n<td>0.01<\/td>\r\n<td>0.999983333417<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.001<\/td>\r\n<td>0.999999833333<\/td>\r\n<td>0.001<\/td>\r\n<td>0.999999833333<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.0001<\/td>\r\n<td>0.999999998333<\/td>\r\n<td>0.0001<\/td>\r\n<td>0.999999998333<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572236179\"><em>Note<\/em>: The values in this table were obtained using a calculator and using all the places given in the calculator output.<\/p>\r\n<p id=\"fs-id1170572558630\">As we read down each [latex]\\frac{\\sin x}{x}[\/latex] column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}=1[\/latex]. A calculator-or computer-generated graph of [latex]f(x)=\\frac{\\sin x}{x}[\/latex] would be similar to that shown in Figure 2, and it confirms our estimate.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202852\/CNX_Calc_Figure_02_02_003.jpg\" alt=\"A graph of f(x) = sin(x)\/x over the interval [-6, 6]. The curving function has a y intercept at x=0 and x intercepts at y=pi and y=-pi.\" width=\"487\" height=\"312\" \/> Figure 2. The graph of [latex]f(x)=(\\sin x)\/x[\/latex] confirms the estimate from the table.[\/caption][\/hidden-answer]<\/div>\r\n<div id=\"fs-id1170571656691\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating a Limit Using a Table of Functional Values 2<\/h3>\r\n<p id=\"fs-id1170572550814\">Evaluate [latex]\\underset{x\\to 4}{\\lim}\\dfrac{\\sqrt{x}-2}{x-4}[\/latex] using a table of functional values.<\/p>\r\n[reveal-answer q=\"fs-id1170572141980\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572141980\"]\r\n<p id=\"fs-id1170572141980\">As before, we use a table to list the values of the function for the given values of [latex]x[\/latex].<\/p>\r\n\r\n<table id=\"fs-id1170571595483\" summary=\"There are two tables, each with six rows and two columns. The first table has headers x and (sqrt(x) \u2013 2 ) \/ (x-4) in the first row. In the first column under x are the values 3.9, 3.99, 3.999, 3.9999, and 3.99999. In the second column are the values 0.251582341869, 0.25015644562, 0.250015627, 0.250001563, 0.25000016. The second table has the same headers in the first row. In the first column under x are the values 4.1, 4.01, 4.001, 4.0001, and 4.00001. In the second column are the values 0.248456731317, 0.24984394501, 0.249984377, 0.249998438, and 0.24999984.\"><caption>Table of Functional Values for [latex]\\underset{x\\to 4}{\\lim}\\frac{\\sqrt{x}-2}{x-4}[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{\\sqrt{x}-2}{x-4}[\/latex]<\/th>\r\n<th><\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{\\sqrt{x}-2}{x-4}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>3.9<\/td>\r\n<td>0.251582341869<\/td>\r\n<td rowspan=\"5\"><\/td>\r\n<td>4.1<\/td>\r\n<td>0.248456731317<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3.99<\/td>\r\n<td>0.25015644562<\/td>\r\n<td>4.01<\/td>\r\n<td>0.24984394501<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3.999<\/td>\r\n<td>0.250015627<\/td>\r\n<td>4.001<\/td>\r\n<td>0.249984377<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3.9999<\/td>\r\n<td>0.250001563<\/td>\r\n<td>4.0001<\/td>\r\n<td>0.249998438<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3.99999<\/td>\r\n<td>0.25000016<\/td>\r\n<td>4.00001<\/td>\r\n<td>0.24999984<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572455426\">After inspecting this table, we see that the functional values less than 4 appear to be decreasing toward 0.25 whereas the functional values greater than 4 appear to be increasing toward 0.25. We conclude that [latex]\\underset{x\\to 4}{\\lim}\\frac{\\sqrt{x}-2}{x-4}=0.25[\/latex]. We confirm this estimate using the graph of [latex]f(x)=\\frac{\\sqrt{x}-2}{x-4}[\/latex] shown in Figure 3.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202855\/CNX_Calc_Figure_02_02_004.jpg\" alt=\"A graph of the function f(x) = (sqrt(x) \u2013 2 ) \/ (x-4) over the interval [0,8]. There is an open circle on the function at x=4. The function curves asymptotically towards the x axis and y axis in quadrant one.\" width=\"487\" height=\"283\" \/> Figure 3. The graph of [latex]f(x)=\\frac{\\sqrt{x}-2}{x-4}[\/latex] confirms the estimate from the table.[\/caption][\/hidden-answer]<\/div>\r\n<div id=\"fs-id1170572212020\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571596043\">Estimate [latex]\\underset{x\\to 1}{\\lim}\\dfrac{\\frac{1}{x}-1}{x-1}[\/latex] using a table of functional values. Use a graph to confirm your estimate.<\/p>\r\n[reveal-answer q=\"377622\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"377622\"]\r\n<p id=\"fs-id1170571656412\">Use 0.9, 0.99, 0.999, 0.9999, 0.99999 and 1.1, 1.01, 1.001, 1.0001, 1.00001 as your table values.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572227899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572227899\"]\r\n<p id=\"fs-id1170572227899\">[latex]\\underset{x\\to 1}{\\lim}\\frac{\\frac{1}{x}-1}{x-1}=-1[\/latex]<\/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]4853[\/ohm_question]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572506486\">At this point, we see from the tables\u00a0that it may be just as easy, if not easier, to estimate a limit of a function by inspecting its graph as it is to estimate the limit by using a table of functional values. In the example below, we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values.\u00a0Recall that looking at a graph, a function's value at a given x value is simply the y value at x.<\/p>\r\n\r\n<div id=\"fs-id1170572337207\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating a Limit Using a Graph<\/h3>\r\n<p id=\"fs-id1170572347401\">For [latex]g(x)[\/latex] shown in Figure 4, evaluate [latex]\\underset{x\\to -1}{\\lim}g(x)[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202858\/CNX_Calc_Figure_02_02_006.jpg\" alt=\"The graph of a generic curving function g(x). In quadrant two, there is an open circle on the function at (-1,3) and a closed circle one unit up at (-1, 4).\" width=\"487\" height=\"390\" \/> Figure 4. The graph of [latex]g(x)[\/latex] includes one value not on a smooth curve.[\/caption][reveal-answer q=\"fs-id1170571654410\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571654410\"]\r\n<p id=\"fs-id1170571654410\">Despite the fact that [latex]g(-1)=4[\/latex], as the [latex]x[\/latex]-values approach \u22121 from either side, the [latex]g(x)[\/latex] values approach 3. Therefore, [latex]\\underset{x\\to -1}{\\lim}g(x)=3[\/latex]. Note that we can determine this limit without even knowing the algebraic expression of the function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170571654758\">Based on the example above, we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different.<\/p>\r\n\r\n\r\n[caption]Watch the following video to see the more examples of evaluating a limit using a graph[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qiHi41CfnFA?controls=0&amp;start=329&amp;end=405&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\/2.2TheLimitOfAFunction329to405_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-id1170571654767\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571656657\">Use the graph of [latex]h(x)[\/latex] in Figure 5 to evaluate [latex]\\underset{x\\to 2}{\\lim}h(x)[\/latex], if possible.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202902\/CNX_Calc_Figure_02_02_007.jpg\" alt=\"A graph of the function h(x), which is a parabola graphed over [-2.5, 5]. There is an open circle where the vertex should be at the point (2,-1).\" width=\"487\" height=\"431\" \/> Figure 5.\u00a0 The graph of [latex]h(x)[\/latex] consists of a smooth graph with a single removed point at [latex]x=2[\/latex].[\/caption][reveal-answer q=\"806443\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"806443\"]\r\n<p id=\"fs-id1170571657959\">What [latex]y[\/latex]-value does the function approach as the [latex]x[\/latex]-values approach 2?<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170571593051\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571593051\"]\r\n<p id=\"fs-id1170571593051\">[latex]\\underset{x\\to 2}{\\lim}h(x)=-1[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572086316\">Looking at a table of functional values or looking at the graph of a function provides us with useful insight into the value of the limit of a function at a given point. However, these techniques rely too much on guesswork. We eventually need to develop alternative methods of evaluating limits. These new methods are more algebraic in nature and we explore them in the next section; however, at this point we introduce two special limits that are foundational to the techniques to come.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Two Important Limits<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572243382\">Let [latex]a[\/latex] be a real number and [latex]c[\/latex] be a constant.<\/p>\r\n\r\n<ol id=\"fs-id1170571659112\">\r\n \t<li>\r\n<div id=\"fs-id1170571611919\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}x=a[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1170571600104\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1170571655925\">We can make the following observations about these two limits.<\/p>\r\n\r\n<ol id=\"fs-id1170572305900\">\r\n \t<li>For the first limit, observe that as [latex]x[\/latex] approaches [latex]a[\/latex], so does [latex]f(x)[\/latex], because [latex]f(x)=x[\/latex]. Consequently, [latex]\\underset{x\\to a}{\\lim}x=a[\/latex].<\/li>\r\n \t<li>For the second limit, consider the table below.<\/li>\r\n<\/ol>\r\n<table id=\"fs-id1170571613026\" summary=\"Two tables side by side, both containing two columns and five rows. The first table has headers x and f(x) = c in the first row. Under x in the first column are the values a-0.1, a-0.01, a-0.001, and a-0.0001. All of the values in the second column under the header are c. The second table has the same headers. Under x in the first column are the values a+0.1, a+0.01, a+0.001, and a+0.0001. All of the values in the second column under the header are c.\"><caption>Table of Functional Values for [latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)=c[\/latex]<\/th>\r\n<th><\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)=c[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.1[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td rowspan=\"4\"><\/td>\r\n<td>[latex]a+0.1[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.01[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]a+0.01[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.001[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]a+0.001[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.0001[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]a+0.0001[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170571576778\">Observe that for all values of [latex]x[\/latex] (regardless of whether they are approaching [latex]a[\/latex]), the values [latex]f(x)[\/latex] remain constant at [latex]c[\/latex]. We have no choice but to conclude [latex]\\underset{x\\to a}{\\lim}c=c[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1170572342287\" class=\"bc-section section\">\r\n<h2>The Existence of a Limit<\/h2>\r\n<p id=\"fs-id1170572342292\">As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.<\/p>\r\n\r\n<div id=\"fs-id1170571656076\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating a Limit That Fails to Exist<\/h3>\r\n<p id=\"fs-id1170571656086\">Evaluate [latex]\\underset{x\\to 0}{\\lim} \\sin \\left(\\dfrac{1}{x}\\right)[\/latex] using a table of values.<\/p>\r\n[reveal-answer q=\"fs-id1170571614817\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571614817\"]\r\n<p id=\"fs-id1170571614817\">The table below lists values for the function [latex] \\sin \\left(\\dfrac{1}{x}\\right)[\/latex] for the given values of [latex]x[\/latex].<\/p>\r\n\r\n<table id=\"fs-id1170572233784\" summary=\"Two tables side by side, each with two columns and seven rows. The headers are the same, x and sin(1\/x) in the first row. In the first table, 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 0.544021110889, 0.50636564111, \u22120;.8268795405312, 0.305614388888, \u22120;.035748797987, and 0.349993504187. 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 \u22120;.544021110889, \u22120;.50636564111, 0.826879540532, \u22120;.305614388888, 0.035748797987, and \u22120;.349993504187.\"><caption>Table of Functional Values for [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex] \\sin (\\frac{1}{x})[\/latex]<\/th>\r\n<th><\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex] \\sin (\\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>0.544021110889<\/td>\r\n<td rowspan=\"6\"><\/td>\r\n<td>0.1<\/td>\r\n<td>\u22120.544021110889<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.01<\/td>\r\n<td>0.50636564111<\/td>\r\n<td>0.01<\/td>\r\n<td>\u22120.50636564111<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.001<\/td>\r\n<td>\u22120.8268795405312<\/td>\r\n<td>0.001<\/td>\r\n<td>0.826879540532<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.0001<\/td>\r\n<td>0.305614388888<\/td>\r\n<td>0.0001<\/td>\r\n<td>\u22120.305614388888<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.00001<\/td>\r\n<td>\u22120.035748797987<\/td>\r\n<td>0.00001<\/td>\r\n<td>0.035748797987<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.000001<\/td>\r\n<td>0.349993504187<\/td>\r\n<td>0.000001<\/td>\r\n<td>\u22120.349993504187<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572420238\">After examining the table of functional values, we can see that the [latex]y[\/latex]-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let\u2019s take a more systematic approach. Take the following sequence of [latex]x[\/latex]-values approaching 0:<\/p>\r\n\r\n<div id=\"fs-id1170572420254\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{2}{\\pi }, \\, \\frac{2}{3\\pi }, \\, \\frac{2}{5\\pi }, \\, \\frac{2}{7\\pi }, \\, \\frac{2}{9\\pi }, \\, \\frac{2}{11\\pi }, \\, \\cdots[\/latex]<\/div>\r\n<p id=\"fs-id1170572561333\">The corresponding [latex]y[\/latex]-values are<\/p>\r\n\r\n<div id=\"fs-id1170572561341\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1, \\, -1, \\, 1, \\, -1, \\, 1, \\, -1, \\, \\cdots[\/latex]<\/div>\r\n<p id=\"fs-id1170571594790\">At this point we can indeed conclude that [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex] does not exist.<\/p>\r\n<em>Mathematicians frequently abbreviate \u201cdoes not exist\u201d as DNE.<\/em>\r\n\r\nThus, we would write [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex] DNE.) The graph of [latex]f(x)= \\sin (\\frac{1}{x})[\/latex] is shown in Figure 6 and it gives a clearer picture of the behavior of [latex] \\sin (\\frac{1}{x})[\/latex] as [latex]x[\/latex] approaches 0. You can see that [latex] \\sin (\\frac{1}{x})[\/latex] oscillates ever more wildly between \u22121 and 1 as [latex]x[\/latex] approaches 0.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img id=\"16\" src=\"https:\/\/openstax.org\/resources\/129955615aaa9011878d34a280fa59baf88b6139\" alt=\"The graph of the function f(x) = sin(1\/x), which oscillates rapidly between -1 and 1 as x approaches 0. The oscillations are less frequent as the function moves away from 0 on the x axis.\" width=\"487\" height=\"358\" data-media-type=\"image\/jpeg\" \/> Figure 6. The graph of [latex]f(x)= \\sin (\\frac{1}{x})[\/latex] oscillates rapidly between \u22121 and 1 as x approaches 0.[\/caption][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572455161\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572455169\">Use a table of functional values to evaluate [latex]\\underset{x\\to 2}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex], if possible.<\/p>\r\n[reveal-answer q=\"338855\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"338855\"]\r\n<p id=\"fs-id1170572560581\">Use [latex]x[\/latex]-values 1.9, 1.99, 1.999, 1.9999, 1.9999 and 2.1, 2.01, 2.001, 2.0001, 2.00001 in your table.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572560593\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572560593\"]\r\n<p id=\"fs-id1170572560593\">[latex]\\underset{x\\to 2}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex] does not exist.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Using correct notation, describe the limit of a function<\/li>\n<li>Use a table of values to estimate the limit of a function or to identify when the limit does not exist<\/li>\n<li>Use a graph to estimate the limit of a function or to identify when the limit does not exist<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1170572450938\">We begin our exploration of limits by taking a look at the graphs of the functions<\/p>\n<div id=\"fs-id1170572346957\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)=\\dfrac{x^2-4}{x-2}, \\ \\, g(x)=\\dfrac{|x-2|}{x-2}[\/latex],\u00a0 and\u00a0 [latex]h(x)=\\dfrac{1}{(x-2)^2}[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572216951\">which are shown in Figure 1. In particular, let\u2019s focus our attention on the behavior of each graph at and around [latex]x=2[\/latex].<\/p>\n<div style=\"width: 985px\" 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\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. These graphs show the behavior of three different functions around [latex]x=2[\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1170572175064\">Each of the three functions is undefined at [latex]x=2[\/latex], but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of [latex]x=2[\/latex]. To express the behavior of each graph in the vicinity of 2 more completely, we need to introduce the concept of a limit.<\/p>\n<div id=\"fs-id1170572280146\" class=\"bc-section section\">\n<h2>Intuitive Definition of a Limit<\/h2>\n<p id=\"fs-id1170572449458\">Let\u2019s first take a closer look at how the function [latex]f(x)=\\dfrac{(x^2-4)}{(x-2)}[\/latex] behaves around [latex]x=2[\/latex] in Figure 1. As the values of [latex]x[\/latex] approach 2 from either side of 2, the values of [latex]y=f(x)[\/latex] approach 4. Mathematically, we say that the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches 2 is 4. Symbolically, we express this limit as<\/p>\n<div id=\"fs-id1170571655049\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x \\to 2}{\\lim}f(x)=4[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572220702\">From this very brief informal look at one limit, let\u2019s start to develop an <strong>intuitive definition of the limit<\/strong>. We can think of the limit of a function at a number [latex]a[\/latex] as being the one real number [latex]L[\/latex] that the functional values approach as the [latex]x[\/latex]-values approach [latex]a[\/latex]<em>,<\/em> provided such a real number [latex]L[\/latex] exists. Stated more carefully, we have the following definition:<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1170572151707\">Let [latex]f(x)[\/latex] be a function defined at all values in an open interval containing [latex]a[\/latex], with the possible exception of [latex]a[\/latex] itself, and let [latex]L[\/latex]\u00a0be a real number. If <em>all<\/em> values of the function [latex]f(x)[\/latex] approach the real number [latex]L[\/latex]\u00a0as the values of [latex]x(\\ne a)[\/latex] approach the number [latex]a[\/latex], then we say that the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is [latex]L[\/latex]. (More succinct, as [latex]x[\/latex] gets closer to [latex]a[\/latex], [latex]f(x)[\/latex] gets closer and stays close to [latex]L[\/latex].) Symbolically, we express this idea as<\/p>\n<div id=\"fs-id1170572133132\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6241\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6241&theme=oea&iframe_resize_id=ohm6241&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p id=\"fs-id1170572244141\">We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy.<\/p>\n<div id=\"fs-id1170571656330\" class=\"textbox examples\">\n<h3>Problem-Solving Strategy: Evaluating a Limit Using a Table of Functional Values<\/h3>\n<ol id=\"fs-id1170572480841\">\n<li>To evaluate [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex], we begin by completing a table of functional values. We should choose two sets of [latex]x[\/latex]-values\u2014one set of values approaching [latex]a[\/latex] and less than [latex]a[\/latex], and another set of values approaching [latex]a[\/latex] and greater than [latex]a[\/latex]. The table below demonstrates what your tables might look like.<br \/>\n<table id=\"fs-id1170572204940\" summary=\"There are two tables. They both have two columns and five rows. The first table has headers x and f(x) in the first row. Under x in the first column are the values a-0.1, a-0.01, a-0.001, and a-0.0001. Under f(x) in the second column are values f(a-0.1), f(a-0.01), f(a-0.001), and f(a-0.0001). At the bottom is a note that one may \u201cuse additional values as necessary\u201d in both columns. The second table has headers x and f(x) in the first row. Under x in the first column are the values a+0.1, a+0.01, a+0.001, and a+0.0001. Under f(x) in the second column are values f(a+0.1), f(a+0.01), f(a+0.001), and f(a+0.0001). At the bottom is a note that one may \u201cuse additional values as necessary\u201d in both columns.\">\n<caption>Table of Functional Values for [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)[\/latex]<\/th>\n<th><\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]a-0.1[\/latex]<\/td>\n<td>[latex]f(a-0.1)[\/latex]<\/td>\n<td rowspan=\"5\"><\/td>\n<td>[latex]a+0.1[\/latex]<\/td>\n<td>[latex]f(a+0.1)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.01[\/latex]<\/td>\n<td>[latex]f(a-0.01)[\/latex]<\/td>\n<td>[latex]a+0.01[\/latex]<\/td>\n<td>[latex]f(a+0.01)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.001[\/latex]<\/td>\n<td>[latex]f(a-0.001)[\/latex]<\/td>\n<td>[latex]a+0.001[\/latex]<\/td>\n<td>[latex]f(a+0.001)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.0001[\/latex]<\/td>\n<td>[latex]f(a-0.0001)[\/latex]<\/td>\n<td>[latex]a+0.0001[\/latex]<\/td>\n<td>[latex]f(a+0.0001)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\">Use additional values as necessary.<\/td>\n<td colspan=\"2\">Use additional values as necessary.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Next, let\u2019s look at the values in each of the [latex]f(x)[\/latex] columns and determine whether the values seem to be approaching a single value as we move down each column. In our columns, we look at the sequence [latex]f(a-0.1), \\, f(a-0.01), \\, f(a-0.001), \\, f(a-0.0001),[\/latex] and so on, and [latex]f(a+0.1), \\, f(a+0.01), \\, f(a+0.001), \\, f(a+0.0001)[\/latex] and so on. (<em>Note<\/em>: Although we have chosen the [latex]x[\/latex]-values [latex]a \\pm 0.1, \\, a \\pm 0.01, \\, a \\pm 0.001, \\, a \\pm 0.0001[\/latex], and so forth, and these values will probably work nearly every time, on very rare occasions we may need to modify our choices.)<\/li>\n<li>If both columns approach a common [latex]y[\/latex]-value [latex]L[\/latex], we state [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]. We can use the following strategy to confirm the result obtained from the table or as an alternative method for estimating a limit.<\/li>\n<li>Using a graphing calculator or computer software that allows us graph functions, we can plot the function [latex]f(x)[\/latex], making sure the functional values of [latex]f(x)[\/latex] for [latex]x[\/latex]-values near [latex]a[\/latex] are in our window. We can use the trace feature to move along the graph of the function and watch the [latex]y[\/latex]-value readout as the [latex]x[\/latex]-values approach [latex]a[\/latex]. If the [latex]y[\/latex]-values approach [latex]L[\/latex]\u00a0as our [latex]x[\/latex]-values approach [latex]a[\/latex] from both directions, then [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]. We may need to zoom in on our graph and repeat this process several times.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1170572175147\">We apply this Problem-Solving Strategy to compute a limit below.<\/p>\n<div id=\"fs-id1170572561451\" class=\"textbook exercises\">\n<h3>Example: Evaluating a Limit Using a Table of Functional Values 1<\/h3>\n<p id=\"fs-id1170571596728\">Evaluate [latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin x}{x}[\/latex] using a table of functional values.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572552454\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572552454\" class=\"hidden-answer\" style=\"display: none\">We have calculated the values of [latex]f(x)=\\dfrac{(\\sin x)}{x}[\/latex] for the values of [latex]x[\/latex] listed in the table below.<\/p>\n<table id=\"fs-id1170572208852\" summary=\"There are two tables. They both have two columns and five rows. The first table has headers x and sin(x)\/x in the first row. Under x in the first column are the values -0.1, -0.01, -0.001, and -0.0001. Under sin(x)\/x in the second column are values 0.998334166468, 0.999983333417, 0.999999833333, and 0.999999998333. The second table has headers x and sin(x)\/x in the first row. Under x in the first column are the values 0.1, 0.01, 0.001, and 0.0001. Under sin(x)\/x in the second column are values 0.998334166468, 0.999983333417, 0.999999833333, and 0.999999998333.\">\n<caption>Table of Functional Values for [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{\\sin x}{x}[\/latex]<\/th>\n<th><\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{\\sin x}{x}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u22120.1<\/td>\n<td>0.998334166468<\/td>\n<td rowspan=\"4\"><\/td>\n<td>0.1<\/td>\n<td>0.998334166468<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.01<\/td>\n<td>0.999983333417<\/td>\n<td>0.01<\/td>\n<td>0.999983333417<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.001<\/td>\n<td>0.999999833333<\/td>\n<td>0.001<\/td>\n<td>0.999999833333<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.0001<\/td>\n<td>0.999999998333<\/td>\n<td>0.0001<\/td>\n<td>0.999999998333<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572236179\"><em>Note<\/em>: The values in this table were obtained using a calculator and using all the places given in the calculator output.<\/p>\n<p id=\"fs-id1170572558630\">As we read down each [latex]\\frac{\\sin x}{x}[\/latex] column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}=1[\/latex]. A calculator-or computer-generated graph of [latex]f(x)=\\frac{\\sin x}{x}[\/latex] would be similar to that shown in Figure 2, and it confirms our estimate.<\/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\/11202852\/CNX_Calc_Figure_02_02_003.jpg\" alt=\"A graph of f(x) = sin(x)\/x over the interval &#091;-6, 6&#093;. The curving function has a y intercept at x=0 and x intercepts at y=pi and y=-pi.\" width=\"487\" height=\"312\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. The graph of [latex]f(x)=(\\sin x)\/x[\/latex] confirms the estimate from the table.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571656691\" class=\"textbook exercises\">\n<h3>Example: Evaluating a Limit Using a Table of Functional Values 2<\/h3>\n<p id=\"fs-id1170572550814\">Evaluate [latex]\\underset{x\\to 4}{\\lim}\\dfrac{\\sqrt{x}-2}{x-4}[\/latex] using a table of functional values.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572141980\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572141980\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572141980\">As before, we use a table to list the values of the function for the given values of [latex]x[\/latex].<\/p>\n<table id=\"fs-id1170571595483\" summary=\"There are two tables, each with six rows and two columns. The first table has headers x and (sqrt(x) \u2013 2 ) \/ (x-4) in the first row. In the first column under x are the values 3.9, 3.99, 3.999, 3.9999, and 3.99999. In the second column are the values 0.251582341869, 0.25015644562, 0.250015627, 0.250001563, 0.25000016. The second table has the same headers in the first row. In the first column under x are the values 4.1, 4.01, 4.001, 4.0001, and 4.00001. In the second column are the values 0.248456731317, 0.24984394501, 0.249984377, 0.249998438, and 0.24999984.\">\n<caption>Table of Functional Values for [latex]\\underset{x\\to 4}{\\lim}\\frac{\\sqrt{x}-2}{x-4}[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{\\sqrt{x}-2}{x-4}[\/latex]<\/th>\n<th><\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{\\sqrt{x}-2}{x-4}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>3.9<\/td>\n<td>0.251582341869<\/td>\n<td rowspan=\"5\"><\/td>\n<td>4.1<\/td>\n<td>0.248456731317<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3.99<\/td>\n<td>0.25015644562<\/td>\n<td>4.01<\/td>\n<td>0.24984394501<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3.999<\/td>\n<td>0.250015627<\/td>\n<td>4.001<\/td>\n<td>0.249984377<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3.9999<\/td>\n<td>0.250001563<\/td>\n<td>4.0001<\/td>\n<td>0.249998438<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3.99999<\/td>\n<td>0.25000016<\/td>\n<td>4.00001<\/td>\n<td>0.24999984<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572455426\">After inspecting this table, we see that the functional values less than 4 appear to be decreasing toward 0.25 whereas the functional values greater than 4 appear to be increasing toward 0.25. We conclude that [latex]\\underset{x\\to 4}{\\lim}\\frac{\\sqrt{x}-2}{x-4}=0.25[\/latex]. We confirm this estimate using the graph of [latex]f(x)=\\frac{\\sqrt{x}-2}{x-4}[\/latex] shown in Figure 3.<\/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\/11202855\/CNX_Calc_Figure_02_02_004.jpg\" alt=\"A graph of the function f(x) = (sqrt(x) \u2013 2 ) \/ (x-4) over the interval &#091;0,8&#093;. There is an open circle on the function at x=4. The function curves asymptotically towards the x axis and y axis in quadrant one.\" width=\"487\" height=\"283\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. The graph of [latex]f(x)=\\frac{\\sqrt{x}-2}{x-4}[\/latex] confirms the estimate from the table.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572212020\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571596043\">Estimate [latex]\\underset{x\\to 1}{\\lim}\\dfrac{\\frac{1}{x}-1}{x-1}[\/latex] using a table of functional values. Use a graph to confirm your estimate.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q377622\">Hint<\/span><\/p>\n<div id=\"q377622\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571656412\">Use 0.9, 0.99, 0.999, 0.9999, 0.99999 and 1.1, 1.01, 1.001, 1.0001, 1.00001 as your table values.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572227899\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572227899\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572227899\">[latex]\\underset{x\\to 1}{\\lim}\\frac{\\frac{1}{x}-1}{x-1}=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm4853\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4853&theme=oea&iframe_resize_id=ohm4853&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p id=\"fs-id1170572506486\">At this point, we see from the tables\u00a0that it may be just as easy, if not easier, to estimate a limit of a function by inspecting its graph as it is to estimate the limit by using a table of functional values. In the example below, we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values.\u00a0Recall that looking at a graph, a function&#8217;s value at a given x value is simply the y value at x.<\/p>\n<div id=\"fs-id1170572337207\" class=\"textbook exercises\">\n<h3>Example: Evaluating a Limit Using a Graph<\/h3>\n<p id=\"fs-id1170572347401\">For [latex]g(x)[\/latex] shown in Figure 4, evaluate [latex]\\underset{x\\to -1}{\\lim}g(x)[\/latex].<\/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\/11202858\/CNX_Calc_Figure_02_02_006.jpg\" alt=\"The graph of a generic curving function g(x). In quadrant two, there is an open circle on the function at (-1,3) and a closed circle one unit up at (-1, 4).\" width=\"487\" height=\"390\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. The graph of [latex]g(x)[\/latex] includes one value not on a smooth curve.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571654410\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571654410\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571654410\">Despite the fact that [latex]g(-1)=4[\/latex], as the [latex]x[\/latex]-values approach \u22121 from either side, the [latex]g(x)[\/latex] values approach 3. Therefore, [latex]\\underset{x\\to -1}{\\lim}g(x)=3[\/latex]. Note that we can determine this limit without even knowing the algebraic expression of the function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170571654758\">Based on the example above, we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different.<\/p>\n<p>Watch the following video to see the more examples of evaluating a limit using a graph<\/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=329&amp;end=405&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\/2.2TheLimitOfAFunction329to405_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-id1170571654767\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571656657\">Use the graph of [latex]h(x)[\/latex] in Figure 5 to evaluate [latex]\\underset{x\\to 2}{\\lim}h(x)[\/latex], if possible.<\/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\/11202902\/CNX_Calc_Figure_02_02_007.jpg\" alt=\"A graph of the function h(x), which is a parabola graphed over [-2.5, 5]. There is an open circle where the vertex should be at the point (2,-1).\" width=\"487\" height=\"431\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5.\u00a0 The graph of [latex]h(x)[\/latex] consists of a smooth graph with a single removed point at [latex]x=2[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q806443\">Hint<\/span><\/p>\n<div id=\"q806443\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571657959\">What [latex]y[\/latex]-value does the function approach as the [latex]x[\/latex]-values approach 2?<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571593051\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571593051\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571593051\">[latex]\\underset{x\\to 2}{\\lim}h(x)=-1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572086316\">Looking at a table of functional values or looking at the graph of a function provides us with useful insight into the value of the limit of a function at a given point. However, these techniques rely too much on guesswork. We eventually need to develop alternative methods of evaluating limits. These new methods are more algebraic in nature and we explore them in the next section; however, at this point we introduce two special limits that are foundational to the techniques to come.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Two Important Limits<\/h3>\n<hr \/>\n<p id=\"fs-id1170572243382\">Let [latex]a[\/latex] be a real number and [latex]c[\/latex] be a constant.<\/p>\n<ol id=\"fs-id1170571659112\">\n<li>\n<div id=\"fs-id1170571611919\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}x=a[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1170571600104\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1170571655925\">We can make the following observations about these two limits.<\/p>\n<ol id=\"fs-id1170572305900\">\n<li>For the first limit, observe that as [latex]x[\/latex] approaches [latex]a[\/latex], so does [latex]f(x)[\/latex], because [latex]f(x)=x[\/latex]. Consequently, [latex]\\underset{x\\to a}{\\lim}x=a[\/latex].<\/li>\n<li>For the second limit, consider the table below.<\/li>\n<\/ol>\n<table id=\"fs-id1170571613026\" summary=\"Two tables side by side, both containing two columns and five rows. The first table has headers x and f(x) = c in the first row. Under x in the first column are the values a-0.1, a-0.01, a-0.001, and a-0.0001. All of the values in the second column under the header are c. The second table has the same headers. Under x in the first column are the values a+0.1, a+0.01, a+0.001, and a+0.0001. All of the values in the second column under the header are c.\">\n<caption>Table of Functional Values for [latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)=c[\/latex]<\/th>\n<th><\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)=c[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]a-0.1[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td rowspan=\"4\"><\/td>\n<td>[latex]a+0.1[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.01[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]a+0.01[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.001[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]a+0.001[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.0001[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]a+0.0001[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170571576778\">Observe that for all values of [latex]x[\/latex] (regardless of whether they are approaching [latex]a[\/latex]), the values [latex]f(x)[\/latex] remain constant at [latex]c[\/latex]. We have no choice but to conclude [latex]\\underset{x\\to a}{\\lim}c=c[\/latex].<\/p>\n<div id=\"fs-id1170572342287\" class=\"bc-section section\">\n<h2>The Existence of a Limit<\/h2>\n<p id=\"fs-id1170572342292\">As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.<\/p>\n<div id=\"fs-id1170571656076\" class=\"textbook exercises\">\n<h3>Example: Evaluating a Limit That Fails to Exist<\/h3>\n<p id=\"fs-id1170571656086\">Evaluate [latex]\\underset{x\\to 0}{\\lim} \\sin \\left(\\dfrac{1}{x}\\right)[\/latex] using a table of values.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571614817\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571614817\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571614817\">The table below lists values for the function [latex]\\sin \\left(\\dfrac{1}{x}\\right)[\/latex] for the given values of [latex]x[\/latex].<\/p>\n<table id=\"fs-id1170572233784\" summary=\"Two tables side by side, each with two columns and seven rows. The headers are the same, x and sin(1\/x) in the first row. In the first table, 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 0.544021110889, 0.50636564111, \u22120;.8268795405312, 0.305614388888, \u22120;.035748797987, and 0.349993504187. 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 \u22120;.544021110889, \u22120;.50636564111, 0.826879540532, \u22120;.305614388888, 0.035748797987, and \u22120;.349993504187.\">\n<caption>Table of Functional Values for [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\sin (\\frac{1}{x})[\/latex]<\/th>\n<th><\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\sin (\\frac{1}{x})[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u22120.1<\/td>\n<td>0.544021110889<\/td>\n<td rowspan=\"6\"><\/td>\n<td>0.1<\/td>\n<td>\u22120.544021110889<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.01<\/td>\n<td>0.50636564111<\/td>\n<td>0.01<\/td>\n<td>\u22120.50636564111<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.001<\/td>\n<td>\u22120.8268795405312<\/td>\n<td>0.001<\/td>\n<td>0.826879540532<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.0001<\/td>\n<td>0.305614388888<\/td>\n<td>0.0001<\/td>\n<td>\u22120.305614388888<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.00001<\/td>\n<td>\u22120.035748797987<\/td>\n<td>0.00001<\/td>\n<td>0.035748797987<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.000001<\/td>\n<td>0.349993504187<\/td>\n<td>0.000001<\/td>\n<td>\u22120.349993504187<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572420238\">After examining the table of functional values, we can see that the [latex]y[\/latex]-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let\u2019s take a more systematic approach. Take the following sequence of [latex]x[\/latex]-values approaching 0:<\/p>\n<div id=\"fs-id1170572420254\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{2}{\\pi }, \\, \\frac{2}{3\\pi }, \\, \\frac{2}{5\\pi }, \\, \\frac{2}{7\\pi }, \\, \\frac{2}{9\\pi }, \\, \\frac{2}{11\\pi }, \\, \\cdots[\/latex]<\/div>\n<p id=\"fs-id1170572561333\">The corresponding [latex]y[\/latex]-values are<\/p>\n<div id=\"fs-id1170572561341\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1, \\, -1, \\, 1, \\, -1, \\, 1, \\, -1, \\, \\cdots[\/latex]<\/div>\n<p id=\"fs-id1170571594790\">At this point we can indeed conclude that [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex] does not exist.<\/p>\n<p><em>Mathematicians frequently abbreviate \u201cdoes not exist\u201d as DNE.<\/em><\/p>\n<p>Thus, we would write [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex] DNE.) The graph of [latex]f(x)= \\sin (\\frac{1}{x})[\/latex] is shown in Figure 6 and it gives a clearer picture of the behavior of [latex]\\sin (\\frac{1}{x})[\/latex] as [latex]x[\/latex] approaches 0. You can see that [latex]\\sin (\\frac{1}{x})[\/latex] oscillates ever more wildly between \u22121 and 1 as [latex]x[\/latex] approaches 0.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" id=\"16\" src=\"https:\/\/openstax.org\/resources\/129955615aaa9011878d34a280fa59baf88b6139\" alt=\"The graph of the function f(x) = sin(1\/x), which oscillates rapidly between -1 and 1 as x approaches 0. The oscillations are less frequent as the function moves away from 0 on the x axis.\" width=\"487\" height=\"358\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. The graph of [latex]f(x)= \\sin (\\frac{1}{x})[\/latex] oscillates rapidly between \u22121 and 1 as x approaches 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572455161\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572455169\">Use a table of functional values to evaluate [latex]\\underset{x\\to 2}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex], if possible.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338855\">Hint<\/span><\/p>\n<div id=\"q338855\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572560581\">Use [latex]x[\/latex]-values 1.9, 1.99, 1.999, 1.9999, 1.9999 and 2.1, 2.01, 2.001, 2.0001, 2.00001 in your table.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572560593\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572560593\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572560593\">[latex]\\underset{x\\to 2}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex] does not exist.<\/p>\n<\/div>\n<\/div>\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-279\">\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":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) 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