{"id":303,"date":"2021-02-04T01:02:33","date_gmt":"2021-02-04T01:02:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=303"},"modified":"2022-03-16T05:22:36","modified_gmt":"2022-03-16T05:22:36","slug":"the-epsilon-delta-definition-of-a-limit","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-epsilon-delta-definition-of-a-limit\/","title":{"raw":"The Epsilon-Delta Definition of a Limit","rendered":"The Epsilon-Delta Definition of a Limit"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Describe the epsilon-delta definition of a limit<\/li>\r\n \t<li>Apply the epsilon-delta definition to find the limit of a function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170571657157\" class=\"bc-section section\">\r\n<h2>Quantifying Closeness<\/h2>\r\n<p id=\"fs-id1170572510272\">Before stating the formal definition of a limit, we must introduce a few preliminary ideas. Recall that the distance between two points [latex]a[\/latex] and [latex]b[\/latex] on a number line is given by [latex]|a-b|[\/latex].<\/p>\r\n\r\n<ul id=\"fs-id1170572408955\">\r\n \t<li>The statement [latex]|f(x)-L|&lt;\\varepsilon [\/latex] may be interpreted as: <em>The distance between [latex]f(x)[\/latex] and [latex]L[\/latex] is less than [latex]\\varepsilon[\/latex].<\/em><\/li>\r\n \t<li>The statement [latex]0&lt;|x-a|&lt;\\delta [\/latex] may be interpreted as: [latex]x\\ne a[\/latex] <em>and the distance between [latex]x[\/latex] and [latex]a[\/latex] is less than [latex]\\delta[\/latex].<\/em><\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170572346904\">It is also important to look at the following equivalences for absolute value:<\/p>\r\n\r\n<ul id=\"fs-id1170571610385\">\r\n \t<li>The statement [latex]|f(x)-L|&lt;\\varepsilon [\/latex] is equivalent to the statement [latex]L-\\varepsilon &lt;f(x)&lt;L+\\varepsilon[\/latex].<\/li>\r\n \t<li>The statement [latex]0&lt;|x-a|&lt;\\delta [\/latex] is equivalent to the statement [latex]a-\\delta &lt;x&lt;a+\\delta [\/latex] and [latex]x\\ne a[\/latex].<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170572552095\">With these clarifications, we can state the formal<strong> epsilon-delta definition of the limit<\/strong>.<\/p>\r\n\r\n<\/div>\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-id1170572482431\">Let [latex]f(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] over an open interval containing [latex]a[\/latex]. Let [latex]L[\/latex] be a real number. Then<\/p>\r\n\r\n<div id=\"fs-id1170571542392\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1170572217764\">if, for every [latex]\\varepsilon &gt;0[\/latex], there exists a [latex]\\delta &gt;0[\/latex] such that if [latex]0&lt;|x-a|&lt;\\delta[\/latex], then [latex]|f(x)-L|&lt;\\varepsilon[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170571657157\" class=\"bc-section section\">\r\n<p id=\"fs-id1170571657966\">This definition may seem rather complex from a mathematical point of view, but it becomes easier to understand if we break it down phrase by phrase. The statement itself involves something called a <span class=\"no-emphasis\"><em>universal quantifier<\/em><\/span> (for every [latex]\\varepsilon &gt;0[\/latex]), an <span class=\"no-emphasis\"><em>existential quantifier<\/em><\/span> (there exists a [latex]\\delta &gt;0[\/latex]), and, last, a <span class=\"no-emphasis\"><em>conditional statement<\/em><\/span> (if [latex]0&lt;|x-a|&lt;\\delta[\/latex] then [latex]|f(x)-L|&lt;\\varepsilon[\/latex]). Let\u2019s take a look at the table below, which breaks down the definition and translates each part.<\/p>\r\n\r\n<table id=\"fs-id1170572305874\" summary=\"A table with two columns and five rows. The first row has the headers \u201cdefinition\u201d and \u201ctranslation,\u201d from mathematical symbols to words. The second row has defines \u201cfor every epsilon less than 0\u201d as \u201cfor every positive distance epsilon from L,\u201d the third row defines \u201cthere exists a delta less than 0\u201d as \u201cthere is a positive difference delta from a,\u201d and the fourth row defines \u201cif 0 is less than the absolute value of x \u2013a, and that is less than delta, then the absolute value of the function of x minus L is less than epsilon\u201d as \u201cif x is closer than delta to a and x is not equal to a, then the function of x is closer than epsilon to L.\u201d\"><caption>Translation of the Epsilon-Delta Definition of the Limit<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Definition<\/th>\r\n<th>Translation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>1. For every [latex]\\varepsilon &gt;0[\/latex],<\/td>\r\n<td>1. For every positive distance [latex]\\varepsilon[\/latex]\u00a0from [latex]L[\/latex],<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>2. there exists a [latex]\\delta &gt;0[\/latex],<\/td>\r\n<td>2. There is a positive distance [latex]\\delta[\/latex] from [latex]a[\/latex],<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>3. such that<\/td>\r\n<td>3. such that<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>4. if [latex]0&lt;|x-a|&lt;\\delta[\/latex], then [latex]|f(x)-L|&lt;\\varepsilon[\/latex].<\/td>\r\n<td>4. if [latex]x[\/latex] is closer than [latex]\\delta [\/latex] to [latex]a[\/latex] and [latex]x\\ne a[\/latex], then [latex]f(x)[\/latex] is closer than [latex]\\varepsilon[\/latex] to [latex]L[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]6242[\/ohm_question]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572236664\">We can get a better handle on this definition by looking at the definition geometrically. Figure 1 shows possible values of [latex]\\delta[\/latex] for various choices of [latex]\\varepsilon &gt;0[\/latex] for a given function [latex]f(x)[\/latex], a number [latex]a[\/latex], and a limit [latex]L[\/latex] at [latex]a[\/latex]. Notice that as we choose smaller values of [latex]\\varepsilon[\/latex]\u00a0(the distance between the function and the limit), we can always find a [latex]\\delta[\/latex] small enough so that if we have chosen an [latex]x[\/latex] value within [latex]\\delta[\/latex] of [latex]a[\/latex], then the value of [latex]f(x)[\/latex] is within [latex]\\varepsilon[\/latex] of the limit [latex]L[\/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\/11203532\/CNX_Calc_Figure_02_05_001.jpg\" alt=\"There are three graphs side by side showing possible values of delta, given successively smaller choices of epsilon. Each graph has a decreasing, concave down curve in quadrant one. Each graph has the point (a, L) marked on the curve, where L is the limit of the function at the point where x=a. On either side of L on the y axis, a distance epsilon is marked off - namely, a line is drawn through the function at y = L + epsilon and L \u2013 epsilon. As smaller values of epsilon are chosen going from graph one to graph three, smaller values of delta to the left and right of point a can be found so that if we have chosen an x value within delta of a, then the value of f(x) is within epsilon of the limit L.\" width=\"975\" height=\"347\" \/> Figure 1. These graphs show possible values of [latex]\\delta[\/latex], given successively smaller choices of [latex]\\varepsilon[\/latex].[\/caption]\r\n<div id=\"fs-id1170571653484\" class=\"textbox tryit\">\r\n<h3>Interactive<\/h3>\r\n<p id=\"fs-id1170572452431\"><a href=\"https:\/\/www.geogebra.org\/m\/FQwxkVbK\" target=\"_blank\" rel=\"noopener\">Visit the following applet to experiment with finding values of [latex]\\delta[\/latex] for selected values of [latex]\\varepsilon[\/latex].<\/a><\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572560080\">The example below shows how you can use this definition to prove a statement about the limit of a specific function at a specified value.<\/p>\r\n\r\n<div id=\"fs-id1170572333023\" class=\"textbook exercises\">\r\n<h3>Example: Proving a Statement about the Limit of a Specific Function<\/h3>\r\n<p id=\"fs-id1170572175004\">Prove that [latex]\\underset{x\\to 1}{\\lim}(2x+1)=3[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170571636491\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571636491\"]\r\n<p id=\"fs-id1170571636491\">Let [latex]\\varepsilon &gt;0[\/latex].<\/p>\r\n<p id=\"fs-id1170572453649\">The first part of the definition begins \u201cFor every [latex]\\varepsilon &gt;0[\/latex].\" This means we must prove that whatever follows is true no matter what positive value of [latex]\\varepsilon[\/latex] is chosen. By stating \u201cLet [latex]\\varepsilon &gt;0[\/latex],\" we signal our intent to do so.<\/p>\r\n<p id=\"fs-id1170572228084\">Choose [latex]\\delta =\\frac{\\varepsilon}{2}[\/latex].<\/p>\r\n<p id=\"fs-id1170572508695\">The definition continues with \u201cthere exists a [latex]\\delta &gt;0[\/latex].\u201d The phrase \u201cthere exists\u201d in a mathematical statement is always a signal for a scavenger hunt. In other words, we must go and find [latex]\\delta[\/latex]. So, where exactly did [latex]\\delta =\\varepsilon\/2[\/latex] come from? There are two basic approaches to tracking down [latex]\\delta[\/latex]. One method is purely algebraic and the other is geometric.<\/p>\r\n<p id=\"fs-id1170572481139\">We begin by tackling the problem from an algebraic point of view. Since ultimately we want [latex]|(2x+1)-3|&lt;\\varepsilon[\/latex], we begin by manipulating this expression: [latex]|(2x+1)-3|&lt;\\varepsilon[\/latex] is equivalent to [latex]|2x-2|&lt;\\varepsilon[\/latex], which in turn is equivalent to [latex]|2||x-1|&lt;\\varepsilon[\/latex]. Last, this is equivalent to [latex]|x-1|&lt;\\varepsilon\/2[\/latex]. Thus, it would seem that [latex]\\delta =\\varepsilon\/2[\/latex] is appropriate.<\/p>\r\n<p id=\"fs-id1170572227894\">We may also find [latex]\\delta [\/latex] through geometric methods. Figure 2 demonstrates how this is done.<\/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\/11203535\/CNX_Calc_Figure_02_05_002.jpg\" alt=\"This graph shows how to find delta geometrically. The function 2x + 1 is drawn in red from x=0 to 2. A straight line is drawn at y=3 in green, which intersects the function at (1,3). Two blues lines are drawn at 3 + epsilon and 3 \u2013 epsilon, which are graphed here between 5 and 6 and between 0 and 1, respectively. Finally, two pink lines are drawn down from the points of intersection of the function and the blue lines \u2013 the taller between 1 and 2, and the shorter between 0 and 1. Since the blue lines and the function intersect, we can solve for x. For the shorter, corresponding to the line y = 3 \u2013 epsilon, we have 3 \u2013 epsilon = 2x + 1, which simplifies to x = 1 \u2013 epsilon \/ 2. For the taller, corresponding to the line y = 3 + epsilon, we have 3 + epsilon = 2x + 1, which simplifies to x = 1 + epsilon \/ 2. Delta is the smaller of the two distances between 1 and where the pink lines intersect with the x axis. We have delta is the min of 1 + epsilon \/ 2 -1 and 1 \u2013 (1 \u2013 epsilon \/ 2), which is the min of epsilon \/ 2 and epsilon \/ 2, which is simply epsilon \/ 2.\" width=\"731\" height=\"430\" \/> Figure 2. This graph shows how we find [latex]\\delta[\/latex] geometrically.[\/caption]\r\n<p id=\"fs-id1170572203818\">Assume [latex]0&lt;|x-1|&lt;\\delta[\/latex]. When [latex]\\delta[\/latex] has been chosen, our goal is to show that if [latex]0&lt;|x-1|&lt;\\delta[\/latex], then [latex]|(2x+1)-3|&lt;\\varepsilon[\/latex]. To prove any statement of the form \u201cIf this, then that,\u201d we begin by assuming \u201cthis\u201d and trying to get \u201cthat.\u201d<\/p>\r\n<p id=\"fs-id1170571616033\">Thus,<\/p>\r\n[latex]\\begin{array}{lllll}|(2x+1)-3| &amp; =|2x-2| &amp; &amp; &amp; \\\\ &amp; =|2(x-1)| \\\\ &amp; =|2||x-1| &amp; &amp; &amp; \\text{property of absolute values:} \\, |ab|=|a||b| \\\\ &amp; =2|x-1| &amp; &amp; &amp; \\\\ &amp; &lt;2 \\cdot \\delta &amp; &amp; &amp; \\text{here\u2019s where we use the assumption that} \\, 0&lt;|x-1|&lt;\\delta \\\\ &amp; =2 \\cdot \\frac{\\varepsilon}{2}=\\varepsilon &amp; &amp; &amp; \\text{here\u2019s where we use our choice of} \\, \\delta =\\varepsilon\/2 \\end{array}[\/latex]\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1170572509826\">In this part of the proof, we started with [latex]|(2x+1)-3|[\/latex] and used our assumption [latex]0&lt;|x-1|&lt;\\delta[\/latex] in a key part of the chain of inequalities to get [latex]|(2x+1)-3|[\/latex] to be less than [latex]\\varepsilon[\/latex]. We could just as easily have manipulated the assumed inequality [latex]0&lt;|x-1|&lt;\\delta[\/latex] to arrive at [latex]|(2x+1)-3|&lt;\\varepsilon[\/latex] as follows:<\/p>\r\n<p id=\"fs-id1170572557581\">[latex]\\begin{array}{ll} 0&lt;|x-1|&lt;\\delta &amp; \\implies |x-1|&lt;\\delta \\\\ &amp; \\implies -\\delta &lt;x-1&lt;\\delta \\\\ &amp; \\implies -\\frac{\\varepsilon}{2}&lt;x-1&lt;\\frac{\\varepsilon}{2} \\\\ &amp; \\implies -\\varepsilon &lt;2x-2&lt;\\varepsilon \\\\ &amp; \\implies |2x-2|&lt;\\varepsilon \\\\ &amp; \\implies |(2x+1)-3|&lt;\\varepsilon \\end{array}[\/latex]<\/p>\r\n<p id=\"fs-id1170572225805\">Therefore, [latex]\\underset{x\\to 1}{\\lim}(2x+1)=3[\/latex]. (Having completed the proof, we state what we have accomplished.)<\/p>\r\n<p id=\"fs-id1170571609264\">After removing all the remarks, here is a final version of the proof:<\/p>\r\n<p id=\"fs-id1170572351959\" style=\"text-align: left;\">Let [latex]\\varepsilon &gt;0[\/latex].<\/p>\r\n<p id=\"fs-id1170572540641\">Choose [latex]\\delta =\\varepsilon\/2[\/latex].<\/p>\r\n<p id=\"fs-id1170571626719\">Assume [latex]0&lt;|x-1|&lt;\\delta[\/latex].<\/p>\r\n<p id=\"fs-id1170572094856\">Thus,<\/p>\r\n<p id=\"fs-id1170571714257\">[latex]\\begin{array}{ll} |(2x+1)-3|&amp; =|2x-2| \\\\ &amp; =|2(x-1)| \\\\ &amp; =|2||x-1| \\\\ &amp; =2|x-1| \\\\ &amp; &lt;2 \\cdot \\delta \\\\ &amp; =2 \\cdot \\frac{\\varepsilon}{2} \\\\ &amp; =\\varepsilon \\end{array}[\/latex]<\/p>\r\n<p id=\"fs-id1170572330866\">Therefore, [latex]\\underset{x\\to 1}{\\lim}(2x+1)=3[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Proving a Statement about the Limit of a Specific Function. [\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/5_q7_Zx26RY?controls=0&amp;start=149&amp;end=335&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\/2.5PreciseDefinitionOfALimit149to335_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.5 Precise Definition of a Limit\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<p id=\"fs-id1170571636238\">The following Problem-Solving Strategy summarizes the type of proof we worked out above.<\/p>\r\n\r\n<div id=\"fs-id1170571636244\" class=\"textbox examples\">\r\n<h3>Problem-Solving Strategy: Proving That [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] for a Specific Function [latex]f(x)[\/latex]<\/h3>\r\n<ol id=\"fs-id1170572388733\">\r\n \t<li>Let\u2019s begin the proof with the following statement: Let [latex]\\varepsilon &gt;0[\/latex].<\/li>\r\n \t<li>Next, we need to obtain a value for [latex]\\delta[\/latex]. After we have obtained this value, we make the following statement, filling in the blank with our choice of [latex]\\delta[\/latex]: Choose [latex]\\delta =[\/latex] _______.<\/li>\r\n \t<li>The next statement in the proof should be (filling in our given value for [latex]a[\/latex]):\r\nAssume [latex]0&lt;|x-a|&lt;\\delta[\/latex].<\/li>\r\n \t<li>Next, based on this assumption, we need to show that [latex]|f(x)-L|&lt;\\varepsilon[\/latex], where [latex]f(x)[\/latex] and [latex]L[\/latex] are our function [latex]f(x)[\/latex] and our limit [latex]L[\/latex]. At some point, we need to use [latex]0&lt;|x-a|&lt;\\delta[\/latex].<\/li>\r\n \t<li>We conclude our proof with the statement: Therefore, [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170571597356\" class=\"textbook exercises\">\r\n<h3>Example: Proving a Statement about a Limit<\/h3>\r\n<p id=\"fs-id1170571597366\">Complete the proof that [latex]\\underset{x\\to -1}{\\lim}(4x+1)=-3[\/latex] by filling in the blanks.<\/p>\r\n<p id=\"fs-id1170572444308\">Let _____.<\/p>\r\n<p id=\"fs-id1170572444311\">Choose [latex]\\delta =[\/latex] ________.<\/p>\r\n<p id=\"fs-id1170572444323\">Assume [latex]0&lt;|x-\\text{___}|&lt;\\delta[\/latex].<\/p>\r\n<p id=\"fs-id1170572560044\">Thus, [latex]|\\text{________}-\\text{___}| =|\\text{_________}| = |\\text{___}||\\text{_________}| = \\text{___} \\, |\\text{_______}| &lt; \\text{______} = \\text{_______} = \\varepsilon[\/latex].<\/p>\r\nTherefore, [latex]\\underset{x \\to -1}{\\lim}(4x+1)=-3[\/latex].\r\n\r\n[reveal-answer q=\"fs-id1170572550132\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572550132\"]\r\n<p id=\"fs-id1170572550132\">We begin by filling in the blanks where the choices are specified by the definition. Thus, we have<\/p>\r\nLet [latex]\\varepsilon &gt;0[\/latex].\r\n<p id=\"fs-id1170571734052\">Choose [latex]\\delta =[\/latex] _______. (Leave this one blank for now -- we'll choose [latex]\\delta[\/latex] later)<\/p>\r\n<p id=\"fs-id1170571734064\">Assume [latex]0&lt;|x-(-1)|&lt;\\delta[\/latex] (or equivalently, [latex]0&lt;|x+1|&lt;\\delta[\/latex]).<\/p>\r\n<p id=\"fs-id1170572626658\">Thus, [latex]|(4x+1)-(-3)|=|4x+4|=|4||x+1|&lt;4\\delta = \\text{_______} = \\varepsilon[\/latex].<\/p>\r\n<p id=\"fs-id1170571609399\">Focusing on the final line of the proof, we see that we should choose [latex]\\delta =\\frac{\\varepsilon}{4}[\/latex].<\/p>\r\n<p id=\"fs-id1170572311242\">We now complete the final write-up of the proof:<\/p>\r\n<p id=\"fs-id1170572311246\">Let [latex]\\varepsilon &gt;0[\/latex].<\/p>\r\n<p id=\"fs-id1170571599525\">Choose [latex]\\delta =\\frac{\\varepsilon}{4}[\/latex].<\/p>\r\n<p id=\"fs-id1170571599542\">Assume [latex]0&lt;|x-(-1)|&lt;\\delta[\/latex] (or equivalently, [latex]0&lt;|x+1|&lt;\\delta[\/latex]).<\/p>\r\n<p id=\"fs-id1170572346800\">Thus, [latex]|(4x+1)-(-3)|=|4x+4|=|4||x+1|&lt;4\\delta =4(\\varepsilon\/4)=\\varepsilon[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1170571712674\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571712682\">Complete the proof that [latex]\\underset{x\\to 2}{\\lim}(3x-2)=4[\/latex] by filling in the blanks.<\/p>\r\n<p id=\"fs-id1170572444308\">Let _____.<\/p>\r\n<p id=\"fs-id1170572444311\">Choose [latex]\\delta =[\/latex] ________.<\/p>\r\n<p id=\"fs-id1170572444323\">Assume [latex]0&lt;|x-\\text{___}|&lt;\\delta[\/latex].<\/p>\r\n<p id=\"fs-id1170572560044\">Thus, [latex]|\\text{________}-\\text{___}| =|\\text{_________}| = |\\text{___}||\\text{_________}| = \\text{___} \\, |\\text{_______}| &lt; \\text{______} = \\text{_______} = \\varepsilon[\/latex].<\/p>\r\n<p id=\"fs-id1170572292953\">Therefore, [latex]\\underset{x\\to 2}{\\lim}(3x-2)=4[\/latex].<\/p>\r\n[reveal-answer q=\"4567890\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"4567890\"]\r\n<p id=\"fs-id1170572448141\">Follow the outline in the Problem-Solving Strategy that we worked out in full in the example above.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170571572023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571572023\"]\r\n<p id=\"fs-id1170571572023\">Let [latex]\\varepsilon &gt;0[\/latex]; choose [latex]\\delta =\\frac{\\varepsilon}{3}[\/latex]; assume [latex]0&lt;|x-2|&lt;\\delta[\/latex].<\/p>\r\n<p id=\"fs-id1170572311214\">Thus, [latex]|(3x-2)-4|=|3x-6|=|3| \\cdot |x-2|&lt;3 \\cdot \\delta =3 \\cdot (\\varepsilon\/3)=\\varepsilon[\/latex].<\/p>\r\nTherefore, [latex]\\underset{x\\to 2}{\\lim}3x-2=4[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170571657104\">In the examples above, the proofs were fairly straightforward, since the functions with which we were working were linear. In the example below, we see how to modify the proof to accommodate a nonlinear function.<\/p>\r\n\r\n<div id=\"fs-id1170571657118\" class=\"textbook exercises\">\r\n<h3>Example: Proving a Statement about the Limit of a Specific Function (Geometric Approach)<\/h3>\r\n<p id=\"fs-id1170571712153\">Prove that [latex]\\underset{x\\to 2}{\\lim}x^2=4[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170571636114\"]Solution[\/reveal-answer][hidden-answer a=\"fs-id1170571636114\"]\r\n<ol id=\"fs-id1170571636114\">\r\n \t<li>Let [latex]\\varepsilon &gt;0[\/latex]. The first part of the definition begins \u201cFor every [latex]\\varepsilon &gt;0[\/latex],\" so we must prove that whatever follows is true no matter what positive value of [latex]\\varepsilon[\/latex] is chosen. By stating \u201cLet [latex]\\varepsilon &gt;0[\/latex],\" we signal our intent to do so.<\/li>\r\n \t<li>Without loss of generality, assume [latex]\\varepsilon \\le 4[\/latex]. Two questions present themselves: Why do we want [latex]\\varepsilon \\le 4[\/latex] and why is it okay to make this assumption? In answer to the first question: Later on, in the process of solving for [latex]\\delta[\/latex], we will discover that [latex]\\delta[\/latex] involves the quantity [latex]\\sqrt{4-\\varepsilon}[\/latex]. Consequently, we need [latex]\\varepsilon \\le 4[\/latex]. In answer to the second question: If we can find [latex]\\delta &gt;0[\/latex] that \u201cworks\u201d for [latex]\\varepsilon \\le 4[\/latex], then it will \u201cwork\u201d for any [latex]\\varepsilon &gt;4[\/latex] as well. Keep in mind that, although it is always okay to put an upper bound on [latex]\\varepsilon[\/latex], it is never okay to put a lower bound (other than zero) on [latex]\\varepsilon[\/latex].<\/li>\r\n \t<li>Choose [latex]\\delta =\\text{min}\\{2-\\sqrt{4-\\varepsilon},\\sqrt{4+\\varepsilon}-2\\}[\/latex]. Figure 3 shows how we made this choice of [latex]\\delta[\/latex].[caption id=\"\" align=\"aligncenter\" width=\"590\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203538\/CNX_Calc_Figure_02_05_003.jpg\" alt=\"This graph shows how to find delta geometrically for a given epsilon for the above proof. First, the function f(x) = x^2 is drawn from [-1, 3]. On the y axis, the proposed limit 4 is marked, and the line y=4 is drawn to intersect with the function at (2,4). For a given epsilon, point 4 + epsilon and 4 \u2013 epsilon are marked on the y axis above and below 4. Blue lines are drawn from these points to intersect with the function, where pink lines are drawn from the point of intersection to the x axis. These lines land on either side of x=2. Next, we solve for these x values, which have to be positive here. The first is x^2 = 4 \u2013 epsilon, which simplifies to x = sqrt(4-epsilon). The next is x^2 = 4 + epsilon, which simplifies to x = sqrt(4 + epsilon). Delta is the smaller of the two distances, so it is the min of (2 \u2013 sqrt(4 \u2013 epsilon) and sqrt(4 + epsilon) \u2013 2).\" width=\"590\" height=\"311\" \/> Figure 3. This graph shows how we find [latex]\\delta[\/latex] geometrically for a given [latex]\\varepsilon[\/latex] for the proof in this example.[\/caption]<\/li>\r\n \t<li>We must show: If [latex]0&lt;|x-2|&lt;\\delta[\/latex], then [latex]|x^2-4|&lt;\\varepsilon[\/latex], so we must begin by assuming\r\n<div id=\"fs-id1170572410134\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]0&lt;|x-2|&lt;\\delta[\/latex].<\/div>\r\nWe don\u2019t really need [latex]0&lt;|x-2|[\/latex] (in other words, [latex]x\\ne 2[\/latex]) for this proof. Since [latex]0&lt;|x-2|&lt;\\delta \\implies |x-2|&lt;\\delta[\/latex], it is okay to drop [latex]0&lt;|x-2|[\/latex].\r\n<div id=\"fs-id1170572505537\" class=\"equation unnumbered\" style=\"text-align: center;\">So, [latex]|x-2|&lt;\\delta[\/latex], which implies [latex]-\\delta &lt;x-2&lt;\\delta[\/latex].<\/div>\r\n<div id=\"fs-id1170572130609\" class=\"equation unnumbered\"><\/div>\r\nRecall that [latex]\\delta =\\text{min}\\{2-\\sqrt{4-\\varepsilon},\\sqrt{4+\\varepsilon}-2\\}[\/latex]. Thus, [latex]\\delta \\le 2-\\sqrt{4-\\varepsilon}[\/latex] and consequently [latex]-(2-\\sqrt{4-\\varepsilon})\\le -\\delta[\/latex]. We also use [latex]\\delta \\le \\sqrt{4+\\varepsilon}-2[\/latex] here. We might ask at this point: Why did we substitute [latex]2-\\sqrt{4-\\varepsilon}[\/latex] for [latex]\\delta [\/latex] on the left-hand side of the inequality and [latex]\\sqrt{4+\\varepsilon}-2[\/latex] on the right-hand side of the inequality? If we look at Figure 3, we see that [latex]2-\\sqrt{4-\\varepsilon}[\/latex] corresponds to the distance on the left of 2 on the [latex]x[\/latex]-axis and [latex]\\sqrt{4+\\varepsilon}-2[\/latex] corresponds to the distance on the right. Thus,\r\n<div id=\"fs-id1170572436257\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]-(2-\\sqrt{4-\\varepsilon})\\le -\\delta &lt;x-2&lt;\\delta \\le \\sqrt{4+\\varepsilon}-2[\/latex].<\/div>\r\nWe simplify the expression on the left:\r\n<div id=\"fs-id1170572444545\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]-2+\\sqrt{4-\\varepsilon}&lt;x-2&lt;\\sqrt{4+\\varepsilon}-2[\/latex].<\/div>\r\nThen, we add 2 to all parts of the inequality:\r\n<div id=\"fs-id1170572550072\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sqrt{4-\\varepsilon}&lt;x&lt;\\sqrt{4+\\varepsilon}[\/latex].<\/div>\r\nWe square all parts of the inequality. It is okay to do so, since all parts of the inequality are positive:\r\n<div id=\"fs-id1170571591429\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]4-\\varepsilon &lt;x^2&lt;4+\\varepsilon[\/latex].<\/div>\r\nWe subtract 4 from all parts of the inequality:\r\n<div id=\"fs-id1170571650014\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]-\\varepsilon &lt;x^2-4&lt;\\varepsilon[\/latex].<\/div>\r\nLast,\r\n<div id=\"fs-id1170571650046\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|x^2-4|&lt;\\varepsilon[\/latex].<\/div><\/li>\r\n \t<li>Therefore,\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\lim}x^2=4[\/latex].<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571571830\">Find [latex]\\delta[\/latex] corresponding to [latex]\\varepsilon &gt;0[\/latex] for a proof that [latex]\\underset{x\\to 9}{\\lim}\\sqrt{x}=3[\/latex].<\/p>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"81872736\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"81872736\"]\r\n\r\n&nbsp;\r\n<p id=\"fs-id1170572512087\">Draw a graph similar to the one in Figure 3.<\/p>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"fs-id1170572512097\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572512097\"]\r\n<p id=\"fs-id1170572512097\">Choose [latex]\\delta =\\text{min}\\{9-(3-\\varepsilon)^2,(3+\\varepsilon)^2-9\\}[\/latex].<\/p>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<p id=\"fs-id1170571690419\">The geometric approach to proving that the limit of a function takes on a specific value works quite well for some functions. Also, the insight into the formal definition of the limit that this method provides is invaluable. However, we may also approach limit proofs from a purely algebraic point of view. In many cases, an algebraic approach may not only provide us with additional insight into the definition, it may prove to be simpler as well. Furthermore, an algebraic approach is the primary tool used in proofs of statements about limits. For the example below, we take on a purely algebraic approach.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170571690430\" class=\"textbook exercises\">\r\n<h3>Example: Proving a Statement about the Limit of a Specific Function (Algebraic Approach)<\/h3>\r\n<p id=\"fs-id1170571690440\">Prove that [latex]\\underset{x\\to -1}{\\lim}(x^2-2x+3)=6[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170572448153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572448153\"]\r\n<p id=\"fs-id1170572448153\">Let\u2019s use our outline from the Problem-Solving Strategy:<\/p>\r\n\r\n<ol id=\"fs-id1170572448157\">\r\n \t<li>Let [latex]\\varepsilon &gt;0[\/latex].<\/li>\r\n \t<li>Choose [latex]\\delta =\\text{min}\\{1,\\varepsilon\/5\\}[\/latex]. This choice of [latex]\\delta[\/latex] may appear odd at first glance, but it was obtained by taking a look at our ultimate desired inequality: [latex]|(x^2-2x+3)-6|&lt;\\varepsilon[\/latex]. This inequality is equivalent to [latex]|x+1|\\cdot |x-3|&lt;\\varepsilon[\/latex]. At this point, the temptation simply to choose [latex]\\delta =\\frac{\\varepsilon}{x-3}[\/latex] is very strong. Unfortunately, our choice of [latex]\\delta[\/latex] must depend on [latex]\\varepsilon[\/latex] only and no other variable. If we can replace [latex]|x-3|[\/latex] by a numerical value, our problem can be resolved. This is the place where assuming [latex]\\delta \\le 1[\/latex] comes into play. The choice of [latex]\\delta \\le 1[\/latex] here is arbitrary. We could have just as easily used any other positive number. In some proofs, greater care in this choice may be necessary. Now, since [latex]\\delta \\le 1[\/latex] and [latex]|x+1|&lt;\\delta \\le 1[\/latex], we are able to show that [latex]|x-3|&lt;5[\/latex]. Consequently, [latex]|x+1| \\cdot |x-3|&lt;|x+1| \\cdot 5[\/latex]. At this point we realize that we also need [latex]\\delta \\le \\varepsilon\/5[\/latex]. Thus, we choose [latex]\\delta =\\text{min}\\{1,\\varepsilon\/5\\}[\/latex].<\/li>\r\n \t<li>Assume [latex]0&lt;|x+1|&lt;\\delta[\/latex]. Thus,\r\n<div id=\"fs-id1170571562629\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|x+1|&lt;1[\/latex] and [latex]|x+1|&lt;\\frac{\\varepsilon}{5}[\/latex]<\/div>\r\nSince [latex]|x+1|&lt;1[\/latex], we may conclude that [latex]-1&lt;x+1&lt;1[\/latex]. Thus, by subtracting 4 from all parts of the inequality, we obtain [latex]-5&lt;x-3&lt;\u22121[\/latex]. Consequently, [latex]|x-3|&lt;5[\/latex]. This gives us\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|(x^2-2x+3)-6|=|x+1| \\cdot |x-3|&lt;\\frac{\\varepsilon}{5} \\cdot 5=\\varepsilon[\/latex]<\/div>\r\nTherefore,\r\n<div id=\"fs-id1170572293364\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to -1}{\\lim}(x^2-2x+3)=6[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Proving a Statement about the Limit of a Specific Function (Algebraic Approach). [\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/5_q7_Zx26RY?controls=0&amp;start=789&amp;end=988&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.5PreciseDefinitionofALimit789to988_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.5 Precise Definition of a Limit\" here (opens in a new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170572243112\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572243120\">Complete the proof that [latex]\\underset{x\\to 1}{\\lim}x^2=1[\/latex].<\/p>\r\n<p id=\"fs-id1170572334818\">Let [latex]\\varepsilon &gt;0[\/latex]; choose [latex]\\delta =\\text{min}\\{1,\\frac{\\varepsilon}{3}\\}[\/latex]; assume [latex]0&lt;|x-1|&lt;\\delta[\/latex].<\/p>\r\n<p id=\"fs-id1170571543218\">Since [latex]|x-1|&lt;1[\/latex], we may conclude that [latex]-1&lt;x-1&lt;1[\/latex]. Thus, [latex]1&lt;x+1&lt;3[\/latex]. Hence, [latex]|x+1|&lt;3[\/latex].<\/p>\r\n[reveal-answer q=\"32772344\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"32772344\"]\r\n<p id=\"fs-id1170571712292\">Use the example above as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170571712301\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571712301\"]\r\n<p id=\"fs-id1170571712301\">[latex]|x^2-1|=|x-1| \\cdot |x+1|&lt;\\varepsilon\/3 \\cdot 3=\\varepsilon [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572512400\">You will find that, in general, the more complex a function, the more likely it is that the algebraic approach is the easiest to apply. The algebraic approach is also more useful in proving statements about limits.<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]162454[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Describe the epsilon-delta definition of a limit<\/li>\n<li>Apply the epsilon-delta definition to find the limit of a function<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170571657157\" class=\"bc-section section\">\n<h2>Quantifying Closeness<\/h2>\n<p id=\"fs-id1170572510272\">Before stating the formal definition of a limit, we must introduce a few preliminary ideas. Recall that the distance between two points [latex]a[\/latex] and [latex]b[\/latex] on a number line is given by [latex]|a-b|[\/latex].<\/p>\n<ul id=\"fs-id1170572408955\">\n<li>The statement [latex]|f(x)-L|<\\varepsilon[\/latex] may be interpreted as: <em>The distance between [latex]f(x)[\/latex] and [latex]L[\/latex] is less than [latex]\\varepsilon[\/latex].<\/em><\/li>\n<li>The statement [latex]0<|x-a|<\\delta[\/latex] may be interpreted as: [latex]x\\ne a[\/latex] <em>and the distance between [latex]x[\/latex] and [latex]a[\/latex] is less than [latex]\\delta[\/latex].<\/em><\/li>\n<\/ul>\n<p id=\"fs-id1170572346904\">It is also important to look at the following equivalences for absolute value:<\/p>\n<ul id=\"fs-id1170571610385\">\n<li>The statement [latex]|f(x)-L|<\\varepsilon[\/latex] is equivalent to the statement [latex]L-\\varepsilon <f(x)<L+\\varepsilon[\/latex].<\/li>\n<li>The statement [latex]0<|x-a|<\\delta[\/latex] is equivalent to the statement [latex]a-\\delta <x<a+\\delta[\/latex] and [latex]x\\ne a[\/latex].<\/li>\n<\/ul>\n<p id=\"fs-id1170572552095\">With these clarifications, we can state the formal<strong> epsilon-delta definition of the limit<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1170572482431\">Let [latex]f(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] over an open interval containing [latex]a[\/latex]. Let [latex]L[\/latex] be a real number. Then<\/p>\n<div id=\"fs-id1170571542392\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1170572217764\">if, for every [latex]\\varepsilon >0[\/latex], there exists a [latex]\\delta >0[\/latex] such that if [latex]0<|x-a|<\\delta[\/latex], then [latex]|f(x)-L|<\\varepsilon[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1170571657157\" class=\"bc-section section\">\n<p id=\"fs-id1170571657966\">This definition may seem rather complex from a mathematical point of view, but it becomes easier to understand if we break it down phrase by phrase. The statement itself involves something called a <span class=\"no-emphasis\"><em>universal quantifier<\/em><\/span> (for every [latex]\\varepsilon >0[\/latex]), an <span class=\"no-emphasis\"><em>existential quantifier<\/em><\/span> (there exists a [latex]\\delta >0[\/latex]), and, last, a <span class=\"no-emphasis\"><em>conditional statement<\/em><\/span> (if [latex]0<|x-a|<\\delta[\/latex] then [latex]|f(x)-L|<\\varepsilon[\/latex]). Let\u2019s take a look at the table below, which breaks down the definition and translates each part.<\/p>\n<table id=\"fs-id1170572305874\" summary=\"A table with two columns and five rows. The first row has the headers \u201cdefinition\u201d and \u201ctranslation,\u201d from mathematical symbols to words. The second row has defines \u201cfor every epsilon less than 0\u201d as \u201cfor every positive distance epsilon from L,\u201d the third row defines \u201cthere exists a delta less than 0\u201d as \u201cthere is a positive difference delta from a,\u201d and the fourth row defines \u201cif 0 is less than the absolute value of x \u2013a, and that is less than delta, then the absolute value of the function of x minus L is less than epsilon\u201d as \u201cif x is closer than delta to a and x is not equal to a, then the function of x is closer than epsilon to L.\u201d\">\n<caption>Translation of the Epsilon-Delta Definition of the Limit<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>Definition<\/th>\n<th>Translation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>1. For every [latex]\\varepsilon >0[\/latex],<\/td>\n<td>1. For every positive distance [latex]\\varepsilon[\/latex]\u00a0from [latex]L[\/latex],<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>2. there exists a [latex]\\delta >0[\/latex],<\/td>\n<td>2. There is a positive distance [latex]\\delta[\/latex] from [latex]a[\/latex],<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>3. such that<\/td>\n<td>3. such that<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>4. if [latex]0<|x-a|<\\delta[\/latex], then [latex]|f(x)-L|<\\varepsilon[\/latex].<\/td>\n<td>4. if [latex]x[\/latex] is closer than [latex]\\delta[\/latex] to [latex]a[\/latex] and [latex]x\\ne a[\/latex], then [latex]f(x)[\/latex] is closer than [latex]\\varepsilon[\/latex] to [latex]L[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6242\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6242&theme=oea&iframe_resize_id=ohm6242&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p id=\"fs-id1170572236664\">We can get a better handle on this definition by looking at the definition geometrically. Figure 1 shows possible values of [latex]\\delta[\/latex] for various choices of [latex]\\varepsilon >0[\/latex] for a given function [latex]f(x)[\/latex], a number [latex]a[\/latex], and a limit [latex]L[\/latex] at [latex]a[\/latex]. Notice that as we choose smaller values of [latex]\\varepsilon[\/latex]\u00a0(the distance between the function and the limit), we can always find a [latex]\\delta[\/latex] small enough so that if we have chosen an [latex]x[\/latex] value within [latex]\\delta[\/latex] of [latex]a[\/latex], then the value of [latex]f(x)[\/latex] is within [latex]\\varepsilon[\/latex] of the limit [latex]L[\/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\/11203532\/CNX_Calc_Figure_02_05_001.jpg\" alt=\"There are three graphs side by side showing possible values of delta, given successively smaller choices of epsilon. Each graph has a decreasing, concave down curve in quadrant one. Each graph has the point (a, L) marked on the curve, where L is the limit of the function at the point where x=a. On either side of L on the y axis, a distance epsilon is marked off - namely, a line is drawn through the function at y = L + epsilon and L \u2013 epsilon. As smaller values of epsilon are chosen going from graph one to graph three, smaller values of delta to the left and right of point a can be found so that if we have chosen an x value within delta of a, then the value of f(x) is within epsilon of the limit L.\" width=\"975\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. These graphs show possible values of [latex]\\delta[\/latex], given successively smaller choices of [latex]\\varepsilon[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1170571653484\" class=\"textbox tryit\">\n<h3>Interactive<\/h3>\n<p id=\"fs-id1170572452431\"><a href=\"https:\/\/www.geogebra.org\/m\/FQwxkVbK\" target=\"_blank\" rel=\"noopener\">Visit the following applet to experiment with finding values of [latex]\\delta[\/latex] for selected values of [latex]\\varepsilon[\/latex].<\/a><\/p>\n<\/div>\n<p id=\"fs-id1170572560080\">The example below shows how you can use this definition to prove a statement about the limit of a specific function at a specified value.<\/p>\n<div id=\"fs-id1170572333023\" class=\"textbook exercises\">\n<h3>Example: Proving a Statement about the Limit of a Specific Function<\/h3>\n<p id=\"fs-id1170572175004\">Prove that [latex]\\underset{x\\to 1}{\\lim}(2x+1)=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571636491\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571636491\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571636491\">Let [latex]\\varepsilon >0[\/latex].<\/p>\n<p id=\"fs-id1170572453649\">The first part of the definition begins \u201cFor every [latex]\\varepsilon >0[\/latex].&#8221; This means we must prove that whatever follows is true no matter what positive value of [latex]\\varepsilon[\/latex] is chosen. By stating \u201cLet [latex]\\varepsilon >0[\/latex],&#8221; we signal our intent to do so.<\/p>\n<p id=\"fs-id1170572228084\">Choose [latex]\\delta =\\frac{\\varepsilon}{2}[\/latex].<\/p>\n<p id=\"fs-id1170572508695\">The definition continues with \u201cthere exists a [latex]\\delta >0[\/latex].\u201d The phrase \u201cthere exists\u201d in a mathematical statement is always a signal for a scavenger hunt. In other words, we must go and find [latex]\\delta[\/latex]. So, where exactly did [latex]\\delta =\\varepsilon\/2[\/latex] come from? There are two basic approaches to tracking down [latex]\\delta[\/latex]. One method is purely algebraic and the other is geometric.<\/p>\n<p id=\"fs-id1170572481139\">We begin by tackling the problem from an algebraic point of view. Since ultimately we want [latex]|(2x+1)-3|<\\varepsilon[\/latex], we begin by manipulating this expression: [latex]|(2x+1)-3|<\\varepsilon[\/latex] is equivalent to [latex]|2x-2|<\\varepsilon[\/latex], which in turn is equivalent to [latex]|2||x-1|<\\varepsilon[\/latex]. Last, this is equivalent to [latex]|x-1|<\\varepsilon\/2[\/latex]. Thus, it would seem that [latex]\\delta =\\varepsilon\/2[\/latex] is appropriate.<\/p>\n<p id=\"fs-id1170572227894\">We may also find [latex]\\delta[\/latex] through geometric methods. Figure 2 demonstrates how this is done.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203535\/CNX_Calc_Figure_02_05_002.jpg\" alt=\"This graph shows how to find delta geometrically. The function 2x + 1 is drawn in red from x=0 to 2. A straight line is drawn at y=3 in green, which intersects the function at (1,3). Two blues lines are drawn at 3 + epsilon and 3 \u2013 epsilon, which are graphed here between 5 and 6 and between 0 and 1, respectively. Finally, two pink lines are drawn down from the points of intersection of the function and the blue lines \u2013 the taller between 1 and 2, and the shorter between 0 and 1. Since the blue lines and the function intersect, we can solve for x. For the shorter, corresponding to the line y = 3 \u2013 epsilon, we have 3 \u2013 epsilon = 2x + 1, which simplifies to x = 1 \u2013 epsilon \/ 2. For the taller, corresponding to the line y = 3 + epsilon, we have 3 + epsilon = 2x + 1, which simplifies to x = 1 + epsilon \/ 2. Delta is the smaller of the two distances between 1 and where the pink lines intersect with the x axis. We have delta is the min of 1 + epsilon \/ 2 -1 and 1 \u2013 (1 \u2013 epsilon \/ 2), which is the min of epsilon \/ 2 and epsilon \/ 2, which is simply epsilon \/ 2.\" width=\"731\" height=\"430\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. This graph shows how we find [latex]\\delta[\/latex] geometrically.<\/p>\n<\/div>\n<p id=\"fs-id1170572203818\">Assume [latex]0<|x-1|<\\delta[\/latex]. When [latex]\\delta[\/latex] has been chosen, our goal is to show that if [latex]0<|x-1|<\\delta[\/latex], then [latex]|(2x+1)-3|<\\varepsilon[\/latex]. To prove any statement of the form \u201cIf this, then that,\u201d we begin by assuming \u201cthis\u201d and trying to get \u201cthat.\u201d<\/p>\n<p id=\"fs-id1170571616033\">Thus,<\/p>\n<p>[latex]\\begin{array}{lllll}|(2x+1)-3| & =|2x-2| & & & \\\\ & =|2(x-1)| \\\\ & =|2||x-1| & & & \\text{property of absolute values:} \\, |ab|=|a||b| \\\\ & =2|x-1| & & & \\\\ & <2 \\cdot \\delta & & & \\text{here\u2019s where we use the assumption that} \\, 0<|x-1|<\\delta \\\\ & =2 \\cdot \\frac{\\varepsilon}{2}=\\varepsilon & & & \\text{here\u2019s where we use our choice of} \\, \\delta =\\varepsilon\/2 \\end{array}[\/latex]\n\n\n<h4>Analysis<\/h4>\n<p id=\"fs-id1170572509826\">In this part of the proof, we started with [latex]|(2x+1)-3|[\/latex] and used our assumption [latex]0<|x-1|<\\delta[\/latex] in a key part of the chain of inequalities to get [latex]|(2x+1)-3|[\/latex] to be less than [latex]\\varepsilon[\/latex]. We could just as easily have manipulated the assumed inequality [latex]0<|x-1|<\\delta[\/latex] to arrive at [latex]|(2x+1)-3|<\\varepsilon[\/latex] as follows:<\/p>\n<p id=\"fs-id1170572557581\">[latex]\\begin{array}{ll} 0<|x-1|<\\delta & \\implies |x-1|<\\delta \\\\ & \\implies -\\delta <x-1<\\delta \\\\ & \\implies -\\frac{\\varepsilon}{2}<x-1<\\frac{\\varepsilon}{2} \\\\ & \\implies -\\varepsilon <2x-2<\\varepsilon \\\\ & \\implies |2x-2|<\\varepsilon \\\\ & \\implies |(2x+1)-3|<\\varepsilon \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1170572225805\">Therefore, [latex]\\underset{x\\to 1}{\\lim}(2x+1)=3[\/latex]. (Having completed the proof, we state what we have accomplished.)<\/p>\n<p id=\"fs-id1170571609264\">After removing all the remarks, here is a final version of the proof:<\/p>\n<p id=\"fs-id1170572351959\" style=\"text-align: left;\">Let [latex]\\varepsilon >0[\/latex].<\/p>\n<p id=\"fs-id1170572540641\">Choose [latex]\\delta =\\varepsilon\/2[\/latex].<\/p>\n<p id=\"fs-id1170571626719\">Assume [latex]0<|x-1|<\\delta[\/latex].<\/p>\n<p id=\"fs-id1170572094856\">Thus,<\/p>\n<p id=\"fs-id1170571714257\">[latex]\\begin{array}{ll} |(2x+1)-3|& =|2x-2| \\\\ & =|2(x-1)| \\\\ & =|2||x-1| \\\\ & =2|x-1| \\\\ & <2 \\cdot \\delta \\\\ & =2 \\cdot \\frac{\\varepsilon}{2} \\\\ & =\\varepsilon \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1170572330866\">Therefore, [latex]\\underset{x\\to 1}{\\lim}(2x+1)=3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Proving a Statement about the Limit of a Specific Function. <\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/5_q7_Zx26RY?controls=0&amp;start=149&amp;end=335&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\/2.5PreciseDefinitionOfALimit149to335_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.5 Precise Definition of a Limit&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<p id=\"fs-id1170571636238\">The following Problem-Solving Strategy summarizes the type of proof we worked out above.<\/p>\n<div id=\"fs-id1170571636244\" class=\"textbox examples\">\n<h3>Problem-Solving Strategy: Proving That [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] for a Specific Function [latex]f(x)[\/latex]<\/h3>\n<ol id=\"fs-id1170572388733\">\n<li>Let\u2019s begin the proof with the following statement: Let [latex]\\varepsilon >0[\/latex].<\/li>\n<li>Next, we need to obtain a value for [latex]\\delta[\/latex]. After we have obtained this value, we make the following statement, filling in the blank with our choice of [latex]\\delta[\/latex]: Choose [latex]\\delta =[\/latex] _______.<\/li>\n<li>The next statement in the proof should be (filling in our given value for [latex]a[\/latex]):<br \/>\nAssume [latex]0<|x-a|<\\delta[\/latex].<\/li>\n<li>Next, based on this assumption, we need to show that [latex]|f(x)-L|<\\varepsilon[\/latex], where [latex]f(x)[\/latex] and [latex]L[\/latex] are our function [latex]f(x)[\/latex] and our limit [latex]L[\/latex]. At some point, we need to use [latex]0<|x-a|<\\delta[\/latex].<\/li>\n<li>We conclude our proof with the statement: Therefore, [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170571597356\" class=\"textbook exercises\">\n<h3>Example: Proving a Statement about a Limit<\/h3>\n<p id=\"fs-id1170571597366\">Complete the proof that [latex]\\underset{x\\to -1}{\\lim}(4x+1)=-3[\/latex] by filling in the blanks.<\/p>\n<p id=\"fs-id1170572444308\">Let _____.<\/p>\n<p id=\"fs-id1170572444311\">Choose [latex]\\delta =[\/latex] ________.<\/p>\n<p id=\"fs-id1170572444323\">Assume [latex]0<|x-\\text{___}|<\\delta[\/latex].<\/p>\n<p id=\"fs-id1170572560044\">Thus, [latex]|\\text{________}-\\text{___}| =|\\text{_________}| = |\\text{___}||\\text{_________}| = \\text{___} \\, |\\text{_______}| < \\text{______} = \\text{_______} = \\varepsilon[\/latex].<\/p>\n<p>Therefore, [latex]\\underset{x \\to -1}{\\lim}(4x+1)=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572550132\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572550132\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572550132\">We begin by filling in the blanks where the choices are specified by the definition. Thus, we have<\/p>\n<p>Let [latex]\\varepsilon >0[\/latex].<\/p>\n<p id=\"fs-id1170571734052\">Choose [latex]\\delta =[\/latex] _______. (Leave this one blank for now &#8212; we&#8217;ll choose [latex]\\delta[\/latex] later)<\/p>\n<p id=\"fs-id1170571734064\">Assume [latex]0<|x-(-1)|<\\delta[\/latex] (or equivalently, [latex]0<|x+1|<\\delta[\/latex]).<\/p>\n<p id=\"fs-id1170572626658\">Thus, [latex]|(4x+1)-(-3)|=|4x+4|=|4||x+1|<4\\delta = \\text{_______} = \\varepsilon[\/latex].<\/p>\n<p id=\"fs-id1170571609399\">Focusing on the final line of the proof, we see that we should choose [latex]\\delta =\\frac{\\varepsilon}{4}[\/latex].<\/p>\n<p id=\"fs-id1170572311242\">We now complete the final write-up of the proof:<\/p>\n<p id=\"fs-id1170572311246\">Let [latex]\\varepsilon >0[\/latex].<\/p>\n<p id=\"fs-id1170571599525\">Choose [latex]\\delta =\\frac{\\varepsilon}{4}[\/latex].<\/p>\n<p id=\"fs-id1170571599542\">Assume [latex]0<|x-(-1)|<\\delta[\/latex] (or equivalently, [latex]0<|x+1|<\\delta[\/latex]).<\/p>\n<p id=\"fs-id1170572346800\">Thus, [latex]|(4x+1)-(-3)|=|4x+4|=|4||x+1|<4\\delta =4(\\varepsilon\/4)=\\varepsilon[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571712674\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571712682\">Complete the proof that [latex]\\underset{x\\to 2}{\\lim}(3x-2)=4[\/latex] by filling in the blanks.<\/p>\n<p id=\"fs-id1170572444308\">Let _____.<\/p>\n<p id=\"fs-id1170572444311\">Choose [latex]\\delta =[\/latex] ________.<\/p>\n<p id=\"fs-id1170572444323\">Assume [latex]0<|x-\\text{___}|<\\delta[\/latex].<\/p>\n<p id=\"fs-id1170572560044\">Thus, [latex]|\\text{________}-\\text{___}| =|\\text{_________}| = |\\text{___}||\\text{_________}| = \\text{___} \\, |\\text{_______}| < \\text{______} = \\text{_______} = \\varepsilon[\/latex].<\/p>\n<p id=\"fs-id1170572292953\">Therefore, [latex]\\underset{x\\to 2}{\\lim}(3x-2)=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4567890\">Hint<\/span><\/p>\n<div id=\"q4567890\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572448141\">Follow the outline in the Problem-Solving Strategy that we worked out in full in the example above.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571572023\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571572023\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571572023\">Let [latex]\\varepsilon >0[\/latex]; choose [latex]\\delta =\\frac{\\varepsilon}{3}[\/latex]; assume [latex]0<|x-2|<\\delta[\/latex].<\/p>\n<p id=\"fs-id1170572311214\">Thus, [latex]|(3x-2)-4|=|3x-6|=|3| \\cdot |x-2|<3 \\cdot \\delta =3 \\cdot (\\varepsilon\/3)=\\varepsilon[\/latex].<\/p>\n<p>Therefore, [latex]\\underset{x\\to 2}{\\lim}3x-2=4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170571657104\">In the examples above, the proofs were fairly straightforward, since the functions with which we were working were linear. In the example below, we see how to modify the proof to accommodate a nonlinear function.<\/p>\n<div id=\"fs-id1170571657118\" class=\"textbook exercises\">\n<h3>Example: Proving a Statement about the Limit of a Specific Function (Geometric Approach)<\/h3>\n<p id=\"fs-id1170571712153\">Prove that [latex]\\underset{x\\to 2}{\\lim}x^2=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571636114\">Solution<\/span><\/p>\n<div id=\"qfs-id1170571636114\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170571636114\">\n<li>Let [latex]\\varepsilon >0[\/latex]. The first part of the definition begins \u201cFor every [latex]\\varepsilon >0[\/latex],&#8221; so we must prove that whatever follows is true no matter what positive value of [latex]\\varepsilon[\/latex] is chosen. By stating \u201cLet [latex]\\varepsilon >0[\/latex],&#8221; we signal our intent to do so.<\/li>\n<li>Without loss of generality, assume [latex]\\varepsilon \\le 4[\/latex]. Two questions present themselves: Why do we want [latex]\\varepsilon \\le 4[\/latex] and why is it okay to make this assumption? In answer to the first question: Later on, in the process of solving for [latex]\\delta[\/latex], we will discover that [latex]\\delta[\/latex] involves the quantity [latex]\\sqrt{4-\\varepsilon}[\/latex]. Consequently, we need [latex]\\varepsilon \\le 4[\/latex]. In answer to the second question: If we can find [latex]\\delta >0[\/latex] that \u201cworks\u201d for [latex]\\varepsilon \\le 4[\/latex], then it will \u201cwork\u201d for any [latex]\\varepsilon >4[\/latex] as well. Keep in mind that, although it is always okay to put an upper bound on [latex]\\varepsilon[\/latex], it is never okay to put a lower bound (other than zero) on [latex]\\varepsilon[\/latex].<\/li>\n<li>Choose [latex]\\delta =\\text{min}\\{2-\\sqrt{4-\\varepsilon},\\sqrt{4+\\varepsilon}-2\\}[\/latex]. Figure 3 shows how we made this choice of [latex]\\delta[\/latex].\n<div style=\"width: 600px\" 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\/11203538\/CNX_Calc_Figure_02_05_003.jpg\" alt=\"This graph shows how to find delta geometrically for a given epsilon for the above proof. First, the function f(x) = x^2 is drawn from &#091;-1, 3&#093;. On the y axis, the proposed limit 4 is marked, and the line y=4 is drawn to intersect with the function at (2,4). For a given epsilon, point 4 + epsilon and 4 \u2013 epsilon are marked on the y axis above and below 4. Blue lines are drawn from these points to intersect with the function, where pink lines are drawn from the point of intersection to the x axis. These lines land on either side of x=2. Next, we solve for these x values, which have to be positive here. The first is x^2 = 4 \u2013 epsilon, which simplifies to x = sqrt(4-epsilon). The next is x^2 = 4 + epsilon, which simplifies to x = sqrt(4 + epsilon). Delta is the smaller of the two distances, so it is the min of (2 \u2013 sqrt(4 \u2013 epsilon) and sqrt(4 + epsilon) \u2013 2).\" width=\"590\" height=\"311\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. This graph shows how we find [latex]\\delta[\/latex] geometrically for a given [latex]\\varepsilon[\/latex] for the proof in this example.<\/p>\n<\/div>\n<\/li>\n<li>We must show: If [latex]0<|x-2|<\\delta[\/latex], then [latex]|x^2-4|<\\varepsilon[\/latex], so we must begin by assuming\n\n\n<div id=\"fs-id1170572410134\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]0<|x-2|<\\delta[\/latex].<\/div>\n<p>We don\u2019t really need [latex]0<|x-2|[\/latex] (in other words, [latex]x\\ne 2[\/latex]) for this proof. Since [latex]0<|x-2|<\\delta \\implies |x-2|<\\delta[\/latex], it is okay to drop [latex]0<|x-2|[\/latex].\n\n\n<div id=\"fs-id1170572505537\" class=\"equation unnumbered\" style=\"text-align: center;\">So, [latex]|x-2|<\\delta[\/latex], which implies [latex]-\\delta <x-2<\\delta[\/latex].<\/div>\n<div id=\"fs-id1170572130609\" class=\"equation unnumbered\"><\/div>\n<p>Recall that [latex]\\delta =\\text{min}\\{2-\\sqrt{4-\\varepsilon},\\sqrt{4+\\varepsilon}-2\\}[\/latex]. Thus, [latex]\\delta \\le 2-\\sqrt{4-\\varepsilon}[\/latex] and consequently [latex]-(2-\\sqrt{4-\\varepsilon})\\le -\\delta[\/latex]. We also use [latex]\\delta \\le \\sqrt{4+\\varepsilon}-2[\/latex] here. We might ask at this point: Why did we substitute [latex]2-\\sqrt{4-\\varepsilon}[\/latex] for [latex]\\delta[\/latex] on the left-hand side of the inequality and [latex]\\sqrt{4+\\varepsilon}-2[\/latex] on the right-hand side of the inequality? If we look at Figure 3, we see that [latex]2-\\sqrt{4-\\varepsilon}[\/latex] corresponds to the distance on the left of 2 on the [latex]x[\/latex]-axis and [latex]\\sqrt{4+\\varepsilon}-2[\/latex] corresponds to the distance on the right. Thus,<\/p>\n<div id=\"fs-id1170572436257\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]-(2-\\sqrt{4-\\varepsilon})\\le -\\delta <x-2<\\delta \\le \\sqrt{4+\\varepsilon}-2[\/latex].<\/div>\n<p>We simplify the expression on the left:<\/p>\n<div id=\"fs-id1170572444545\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]-2+\\sqrt{4-\\varepsilon}<x-2<\\sqrt{4+\\varepsilon}-2[\/latex].<\/div>\n<p>Then, we add 2 to all parts of the inequality:<\/p>\n<div id=\"fs-id1170572550072\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sqrt{4-\\varepsilon}<x<\\sqrt{4+\\varepsilon}[\/latex].<\/div>\n<p>We square all parts of the inequality. It is okay to do so, since all parts of the inequality are positive:<\/p>\n<div id=\"fs-id1170571591429\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]4-\\varepsilon <x^2<4+\\varepsilon[\/latex].<\/div>\n<p>We subtract 4 from all parts of the inequality:<\/p>\n<div id=\"fs-id1170571650014\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]-\\varepsilon <x^2-4<\\varepsilon[\/latex].<\/div>\n<p>Last,<\/p>\n<div id=\"fs-id1170571650046\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|x^2-4|<\\varepsilon[\/latex].<\/div>\n<\/li>\n<li>Therefore,\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\lim}x^2=4[\/latex].<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571571830\">Find [latex]\\delta[\/latex] corresponding to [latex]\\varepsilon >0[\/latex] for a proof that [latex]\\underset{x\\to 9}{\\lim}\\sqrt{x}=3[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q81872736\">Hint<\/span><\/p>\n<div id=\"q81872736\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572512087\">Draw a graph similar to the one in Figure 3.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572512097\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572512097\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572512097\">Choose [latex]\\delta =\\text{min}\\{9-(3-\\varepsilon)^2,(3+\\varepsilon)^2-9\\}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<p id=\"fs-id1170571690419\">The geometric approach to proving that the limit of a function takes on a specific value works quite well for some functions. Also, the insight into the formal definition of the limit that this method provides is invaluable. However, we may also approach limit proofs from a purely algebraic point of view. In many cases, an algebraic approach may not only provide us with additional insight into the definition, it may prove to be simpler as well. Furthermore, an algebraic approach is the primary tool used in proofs of statements about limits. For the example below, we take on a purely algebraic approach.<\/p>\n<\/div>\n<div id=\"fs-id1170571690430\" class=\"textbook exercises\">\n<h3>Example: Proving a Statement about the Limit of a Specific Function (Algebraic Approach)<\/h3>\n<p id=\"fs-id1170571690440\">Prove that [latex]\\underset{x\\to -1}{\\lim}(x^2-2x+3)=6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572448153\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572448153\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572448153\">Let\u2019s use our outline from the Problem-Solving Strategy:<\/p>\n<ol id=\"fs-id1170572448157\">\n<li>Let [latex]\\varepsilon >0[\/latex].<\/li>\n<li>Choose [latex]\\delta =\\text{min}\\{1,\\varepsilon\/5\\}[\/latex]. This choice of [latex]\\delta[\/latex] may appear odd at first glance, but it was obtained by taking a look at our ultimate desired inequality: [latex]|(x^2-2x+3)-6|<\\varepsilon[\/latex]. This inequality is equivalent to [latex]|x+1|\\cdot |x-3|<\\varepsilon[\/latex]. At this point, the temptation simply to choose [latex]\\delta =\\frac{\\varepsilon}{x-3}[\/latex] is very strong. Unfortunately, our choice of [latex]\\delta[\/latex] must depend on [latex]\\varepsilon[\/latex] only and no other variable. If we can replace [latex]|x-3|[\/latex] by a numerical value, our problem can be resolved. This is the place where assuming [latex]\\delta \\le 1[\/latex] comes into play. The choice of [latex]\\delta \\le 1[\/latex] here is arbitrary. We could have just as easily used any other positive number. In some proofs, greater care in this choice may be necessary. Now, since [latex]\\delta \\le 1[\/latex] and [latex]|x+1|<\\delta \\le 1[\/latex], we are able to show that [latex]|x-3|<5[\/latex]. Consequently, [latex]|x+1| \\cdot |x-3|<|x+1| \\cdot 5[\/latex]. At this point we realize that we also need [latex]\\delta \\le \\varepsilon\/5[\/latex]. Thus, we choose [latex]\\delta =\\text{min}\\{1,\\varepsilon\/5\\}[\/latex].<\/li>\n<li>Assume [latex]0<|x+1|<\\delta[\/latex]. Thus,\n\n\n<div id=\"fs-id1170571562629\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|x+1|<1[\/latex] and [latex]|x+1|<\\frac{\\varepsilon}{5}[\/latex]<\/div>\n<p>Since [latex]|x+1|<1[\/latex], we may conclude that [latex]-1<x+1<1[\/latex]. Thus, by subtracting 4 from all parts of the inequality, we obtain [latex]-5<x-3<\u22121[\/latex]. Consequently, [latex]|x-3|<5[\/latex]. This gives us\n\n\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]|(x^2-2x+3)-6|=|x+1| \\cdot |x-3|<\\frac{\\varepsilon}{5} \\cdot 5=\\varepsilon[\/latex]<\/div>\n<p>Therefore,<\/p>\n<div id=\"fs-id1170572293364\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to -1}{\\lim}(x^2-2x+3)=6[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Proving a Statement about the Limit of a Specific Function (Algebraic Approach). <\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/5_q7_Zx26RY?controls=0&amp;start=789&amp;end=988&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.5PreciseDefinitionofALimit789to988_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.5 Precise Definition of a Limit&#8221; here (opens in a new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170572243112\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572243120\">Complete the proof that [latex]\\underset{x\\to 1}{\\lim}x^2=1[\/latex].<\/p>\n<p id=\"fs-id1170572334818\">Let [latex]\\varepsilon >0[\/latex]; choose [latex]\\delta =\\text{min}\\{1,\\frac{\\varepsilon}{3}\\}[\/latex]; assume [latex]0<|x-1|<\\delta[\/latex].<\/p>\n<p id=\"fs-id1170571543218\">Since [latex]|x-1|<1[\/latex], we may conclude that [latex]-1<x-1<1[\/latex]. Thus, [latex]1<x+1<3[\/latex]. Hence, [latex]|x+1|<3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q32772344\">Hint<\/span><\/p>\n<div id=\"q32772344\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571712292\">Use the example above as a guide.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571712301\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571712301\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571712301\">[latex]|x^2-1|=|x-1| \\cdot |x+1|<\\varepsilon\/3 \\cdot 3=\\varepsilon[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572512400\">You will find that, in general, the more complex a function, the more likely it is that the algebraic approach is the easiest to apply. The algebraic approach is also more useful in proving statements about limits.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm162454\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=162454&theme=oea&iframe_resize_id=ohm162454&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-303\">\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>2.5 Precise Definition of a Limit. <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":21,"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\":\"2.5 Precise Definition of a Limit\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-303","chapter","type-chapter","status-publish","hentry"],"part":28,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/303","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":25,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/303\/revisions"}],"predecessor-version":[{"id":4797,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/303\/revisions\/4797"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/28"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/303\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=303"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=303"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=303"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=303"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}