{"id":192,"date":"2023-09-20T22:48:10","date_gmt":"2023-09-20T22:48:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/definition-of-a-limit\/"},"modified":"2024-08-05T12:20:28","modified_gmt":"2024-08-05T12:20:28","slug":"introduction-to-the-limit-of-a-function-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-the-limit-of-a-function-learn-it-1\/","title":{"raw":"Introduction to the Limit of a Function: Learn It 1","rendered":"Introduction to the Limit of a Function: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand how to write the limit of a function using the correct symbols and estimate limits by examining tables and graphs<\/li>\r\n\t<li>Understand one-sided limits (approaching a point from only one direction) and how they relate to two-sided limits<\/li>\r\n\t<li>Understand and use the proper notation for infinite limits and define vertical asymptotes<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Definition of a Limit<\/h2>\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<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<p>&nbsp;<\/p>\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\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 [latex]2[\/latex] more completely, we need to introduce the concept of a limit.<\/p>\r\n<div id=\"fs-id1170572280146\" class=\"bc-section section\">\r\n<h3>Intuitive Definition of a Limit<\/h3>\r\n<p id=\"fs-id1170572449458\">Let's examine how the function [latex]f(x)=\\dfrac{(x^2-4)}{(x-2)}[\/latex] behaves as f [latex]x[\/latex] approachs [latex]2[\/latex]. While [latex]f(x)[\/latex] isn't defined at [latex]x=2[\/latex], as [latex]x[\/latex] gets closer to [latex]2[\/latex] from either side, [latex]f(x)[\/latex] approaches [latex]4[\/latex].<\/p>\r\n<p>Mathematically, we say that the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]2[\/latex] is [latex]4[\/latex]. We express this observation using limit notation as:<\/p>\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<p>&nbsp;<\/p>\r\n<p>This initial exploration into limits leads us to a more formal definition.<\/p>\r\n<p>Consider the limit of a function at a specific point as the value that the function's output gets closer to, as the input values approach that point. Assuming such a value exists, we can articulate this concept more precisely with the following definition:<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>limit definition<\/h3>\r\n<p>For a function [latex]f(x)[\/latex] defined over an open interval around a point [latex]a[\/latex], possibly excluding [latex]a[\/latex] itself, if all function values [latex]f(x)[\/latex] get arbitrarily close to some real number [latex]L[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex], then [latex]L[\/latex] is the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex]. <br \/>\r\n<br \/>\r\n<\/p>\r\n<div id=\"fs-id1170572133132\" class=\"equation\" style=\"text-align: center;\">[latex]\\text{Limit Notation: } \\underset{x\\to a}{\\lim}f(x)=L[\/latex]<\/div>\r\n<\/section>\r\n<div>\r\n<section class=\"textbox proTip\">\r\n<p>A more succinct way to understand this definition: As [latex]x[\/latex] gets closer to [latex]a[\/latex], [latex]f(x)[\/latex] gets closer and stays close to [latex]L[\/latex].\u00a0<\/p>\r\n<\/section>\r\n<\/div>\r\n<div>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]6241[\/ohm_question]<\/p>\r\n<\/section>\r\n<h3>Estimating Limits Using Tables\u00a0<\/h3>\r\n<p>We can estimate limits by constructing tables of functional values.\u00a0<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How to: Evaluate a Limit Using a Table of Functional Values<\/strong><\/p>\r\n<ol id=\"fs-id1170572480841\">\r\n\t<li>To find [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex], create a table with two sets of [latex]x[\/latex]-values: those just less than [latex]a[\/latex] and those just more than [latex]a[\/latex]. The table below demonstrates what your tables might look like.<br \/>\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.\">\r\n<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>\u00a0<\/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\">\u00a0<\/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>Analyze the [latex]f(x)[\/latex] values. If they get closer to a single number as [latex]x[\/latex] approaches [latex]a[\/latex] from both sides, that's the limit.<\/li>\r\n\t<li>If both sides of [latex]f(x)[\/latex] is confirmed. If not, the limit may not exist.<\/li>\r\n\t<li>Use the graph of [latex]f(x)[\/latex] to verify your results. By plotting the function and zooming in around [latex]x=a[\/latex], you can observe if [latex]f(x)[\/latex] approaches the limit you calculated. This visual check complements the numerical approach.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\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<p>[reveal-answer q=\"fs-id1170572141980\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572141980\"]<\/p>\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<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.\">\r\n<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>\u00a0<\/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>[latex]3.9[\/latex]<\/td>\r\n<td>[latex]0.251582341869[\/latex]<\/td>\r\n<td rowspan=\"5\">\u00a0<\/td>\r\n<td>[latex]4.1[\/latex]<\/td>\r\n<td>[latex]0.248456731317[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]3.99[\/latex]<\/td>\r\n<td>[latex]0.25015644562[\/latex]<\/td>\r\n<td>[latex]4.01[\/latex]<\/td>\r\n<td>[latex]0.24984394501[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]3.999[\/latex]<\/td>\r\n<td>[latex]0.250015627[\/latex]<\/td>\r\n<td>[latex]4.001[\/latex]<\/td>\r\n<td>[latex]0.249984377[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]3.9999[\/latex]<\/td>\r\n<td>[latex]0.250001563[\/latex]<\/td>\r\n<td>[latex]4.0001[\/latex]<\/td>\r\n<td>[latex]0.249998438[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]3.99999[\/latex]<\/td>\r\n<td>[latex]0.25000016[\/latex]<\/td>\r\n<td>[latex]4.00001[\/latex]<\/td>\r\n<td>[latex]0.24999984[\/latex]<\/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 [latex]4[\/latex] appear to be decreasing toward [latex]0.25[\/latex] whereas the functional values greater than [latex]4[\/latex] appear to be increasing toward [latex]0.25[\/latex]. 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]\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]4853[\/ohm_question]<\/p>\r\n<\/section>\r\n<\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand how to write the limit of a function using the correct symbols and estimate limits by examining tables and graphs<\/li>\n<li>Understand one-sided limits (approaching a point from only one direction) and how they relate to two-sided limits<\/li>\n<li>Understand and use the proper notation for infinite limits and define vertical asymptotes<\/li>\n<\/ul>\n<\/section>\n<h2>The Definition of a Limit<\/h2>\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<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 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 [latex]2[\/latex] more completely, we need to introduce the concept of a limit.<\/p>\n<div id=\"fs-id1170572280146\" class=\"bc-section section\">\n<h3>Intuitive Definition of a Limit<\/h3>\n<p id=\"fs-id1170572449458\">Let&#8217;s examine how the function [latex]f(x)=\\dfrac{(x^2-4)}{(x-2)}[\/latex] behaves as f [latex]x[\/latex] approachs [latex]2[\/latex]. While [latex]f(x)[\/latex] isn&#8217;t defined at [latex]x=2[\/latex], as [latex]x[\/latex] gets closer to [latex]2[\/latex] from either side, [latex]f(x)[\/latex] approaches [latex]4[\/latex].<\/p>\n<p>Mathematically, we say that the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]2[\/latex] is [latex]4[\/latex]. We express this observation using limit notation 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>This initial exploration into limits leads us to a more formal definition.<\/p>\n<p>Consider the limit of a function at a specific point as the value that the function&#8217;s output gets closer to, as the input values approach that point. Assuming such a value exists, we can articulate this concept more precisely with the following definition:<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>limit definition<\/h3>\n<p>For a function [latex]f(x)[\/latex] defined over an open interval around a point [latex]a[\/latex], possibly excluding [latex]a[\/latex] itself, if all function values [latex]f(x)[\/latex] get arbitrarily close to some real number [latex]L[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex], then [latex]L[\/latex] is the limit of [latex]f(x)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex]. <\/p>\n<div id=\"fs-id1170572133132\" class=\"equation\" style=\"text-align: center;\">[latex]\\text{Limit Notation: } \\underset{x\\to a}{\\lim}f(x)=L[\/latex]<\/div>\n<\/section>\n<div>\n<section class=\"textbox proTip\">\n<p>A more succinct way to understand this definition: As [latex]x[\/latex] gets closer to [latex]a[\/latex], [latex]f(x)[\/latex] gets closer and stays close to [latex]L[\/latex].\u00a0<\/p>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm6241\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6241&theme=lumen&iframe_resize_id=ohm6241&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h3>Estimating Limits Using Tables\u00a0<\/h3>\n<p>We can estimate limits by constructing tables of functional values.\u00a0<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Evaluate a Limit Using a Table of Functional Values<\/strong><\/p>\n<ol id=\"fs-id1170572480841\">\n<li>To find [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex], create a table with two sets of [latex]x[\/latex]-values: those just less than [latex]a[\/latex] and those just more 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>\u00a0<\/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\">\u00a0<\/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>Analyze the [latex]f(x)[\/latex] values. If they get closer to a single number as [latex]x[\/latex] approaches [latex]a[\/latex] from both sides, that&#8217;s the limit.<\/li>\n<li>If both sides of [latex]f(x)[\/latex] is confirmed. If not, the limit may not exist.<\/li>\n<li>Use the graph of [latex]f(x)[\/latex] to verify your results. By plotting the function and zooming in around [latex]x=a[\/latex], you can observe if [latex]f(x)[\/latex] approaches the limit you calculated. This visual check complements the numerical approach.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572141980\">Show Solution<\/button><\/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>\u00a0<\/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>[latex]3.9[\/latex]<\/td>\n<td>[latex]0.251582341869[\/latex]<\/td>\n<td rowspan=\"5\">\u00a0<\/td>\n<td>[latex]4.1[\/latex]<\/td>\n<td>[latex]0.248456731317[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]3.99[\/latex]<\/td>\n<td>[latex]0.25015644562[\/latex]<\/td>\n<td>[latex]4.01[\/latex]<\/td>\n<td>[latex]0.24984394501[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]3.999[\/latex]<\/td>\n<td>[latex]0.250015627[\/latex]<\/td>\n<td>[latex]4.001[\/latex]<\/td>\n<td>[latex]0.249984377[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]3.9999[\/latex]<\/td>\n<td>[latex]0.250001563[\/latex]<\/td>\n<td>[latex]4.0001[\/latex]<\/td>\n<td>[latex]0.249998438[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]3.99999[\/latex]<\/td>\n<td>[latex]0.25000016[\/latex]<\/td>\n<td>[latex]4.00001[\/latex]<\/td>\n<td>[latex]0.24999984[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572455426\">After inspecting this table, we see that the functional values less than [latex]4[\/latex] appear to be decreasing toward [latex]0.25[\/latex] whereas the functional values greater than [latex]4[\/latex] appear to be increasing toward [latex]0.25[\/latex]. 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<figure style=\"width: 487px\" 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\" \/><figcaption 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.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm4853\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4853&theme=lumen&iframe_resize_id=ohm4853&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"author":6,"menu_order":11,"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\/192"}],"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":21,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/192\/revisions"}],"predecessor-version":[{"id":4462,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/192\/revisions\/4462"}],"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\/192\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=192"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=192"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=192"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=192"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}