e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
371 lines
15 KiB
HTML
371 lines
15 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/calculus/limits-evaluating.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:48:57 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
|
||
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Limits - Evaluating</title>
|
||
|
||
<style>
|
||
.lim {
|
||
display: inline-table;
|
||
text-align: center;
|
||
vertical-align: middle;
|
||
margin: 0 4px 0 2px;
|
||
border-collapse: collapse;
|
||
}
|
||
.lim em {
|
||
display: table-row;
|
||
text-align: center;
|
||
font-style: inherit;
|
||
}
|
||
.lim strong {
|
||
display: table-row;
|
||
text-align: center;
|
||
font-weight: inherit;
|
||
font-size: 80%;
|
||
line-height: 9px;
|
||
}
|
||
</style>
|
||
|
||
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js"></script>
|
||
<script>document.write(gTagHTML())</script>
|
||
</head>
|
||
|
||
<body id="bodybg" class="adv">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
<h1 class="center">Limits <i>(Evaluating)</i></h1>
|
||
|
||
<p class="center"><i>You should read <a href="limits.html">Limits (An Introduction)</a> first</i></p>
|
||
<h2>Quick Summary of Limits</h2>
|
||
<p>Sometimes we can't work something out directly ... but we <b>can</b> see what it should be as we get closer and closer!</p>
|
||
<div class="example">
|
||
<h3>Example:</h3>
|
||
<p class="center larger"><span class="intbl">
|
||
<em>(x<sup>2</sup> − 1)</em>
|
||
<strong>(x − 1)</strong>
|
||
</span></p>
|
||
<p>Let's work it out for x=1:</p>
|
||
<p class="center larger"><span class="intbl">
|
||
<em>(1<sup>2 </sup>− 1)</em>
|
||
<strong>(1 − 1)</strong>
|
||
</span> = <span class="intbl">
|
||
<em>(1 − 1)</em>
|
||
<strong>(1 − 1)</strong>
|
||
</span> = <span class="intbl">
|
||
<em>0</em>
|
||
<strong>0</strong>
|
||
</span></p>
|
||
</div>
|
||
<p>Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is "indeterminate"), so we need another way of answering this.</p>
|
||
<p>So instead of trying to work it out for x=1 let's try <b>approaching</b> it closer and closer:</p>
|
||
<div class="example">
|
||
<h3>Example Continued:</h3>
|
||
<div class="beach">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:right;">
|
||
<td class="larger">x</td>
|
||
<td style="width:30px;"> </td>
|
||
<td class="larger"><span class="intbl">
|
||
<em>(x<sup>2</sup> − 1)</em>
|
||
<strong>(x − 1)</strong>
|
||
</span></td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td>0.5</td>
|
||
<td> </td>
|
||
<td>1.50000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td>0.9</td>
|
||
<td> </td>
|
||
<td>1.90000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td>0.99</td>
|
||
<td> </td>
|
||
<td>1.99000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td>0.999</td>
|
||
<td> </td>
|
||
<td>1.99900</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td>0.9999</td>
|
||
<td> </td>
|
||
<td>1.99990</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td>0.99999</td>
|
||
<td> </td>
|
||
<td>1.99999</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td>...</td>
|
||
<td> </td>
|
||
<td>...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>Now we see that as x gets close to 1, then <span class="intbl">
|
||
<em>(x<sup>2</sup>−1)</em>
|
||
<strong>(x−1)</strong>
|
||
</span> gets <b>close to 2</b></p>
|
||
</div>
|
||
<p>We are now faced with an interesting situation:</p>
|
||
<ul>
|
||
<li>When x=1 we don't know the answer (it is <b>indeterminate</b>)</li>
|
||
<li>But we can see that it is <b>going to be 2</b></li>
|
||
</ul>
|
||
<p>We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit"</p>
|
||
<p class="center larger">The <b>limit</b> of <span class="intbl">
|
||
<em>(x<sup>2</sup>−1)</em>
|
||
<strong>(x−1)</strong>
|
||
</span> as x approaches 1 is<b> 2</b></p>
|
||
<p>And it is written in symbols as:</p>
|
||
<p class="center large"><span class="lim"><em>lim</em><strong>x→1</strong></span><span class="intbl"><em>x<sup>2</sup>−1</em><strong>x−1</strong></span> = 2</p>
|
||
<p>So it is a special way of saying,<i> "ignoring what happens when we get there, but as we get closer and closer the answer gets closer and closer to 2"</i></p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;">
|
||
<p>As a graph it looks like this:</p>
|
||
<p>So, in truth, we <b>cannot say what the value at x=1 is.</b></p>
|
||
<p>But we <b>can</b> say that as we approach 1, <b>the limit is 2.</b></p>
|
||
</td>
|
||
<td style="text-align:right;"> </td>
|
||
<td><img src="images/graph-x2-1-x-1.svg" alt="graph hole"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<h2>Evaluating Limits</h2>
|
||
<p>"Evaluating" means to find the value of (<i>think e-"<b>value"</b>-ating</i>)</p>
|
||
<p>In the example above we said the limit was 2 because it <b>looked like it was going to be</b>. But that is not really good enough!</p>
|
||
<p>In fact there are <b>many ways</b> to get an accurate answer. Let's look at some:</p>
|
||
<p> </p>
|
||
<h3>1. Just Put The Value In</h3>
|
||
<p>The first thing to try is just putting the value of the limit in, and see if it works (in other words <a href="../algebra/substitution.html">substitution</a>).</p>
|
||
<div class="example">
|
||
<h3>Example:</h3>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<td class="larger"><span class="lim"><em>lim</em><strong>x→10</strong></span><span class="intbl"><em>x</em><strong>2</strong></span></td><!-- limx->10 x/2 -->
|
||
<td> <img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"> </td>
|
||
<td class="larger"><span class="intbl"><em>10</em><strong>2</strong></span> = 5</td>
|
||
<td> <img src="../images/style/yes.svg" alt="yes"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>Easy!</p>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example:</h3>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<td class="larger"><span class="lim"><em>lim</em><strong>x→1</strong></span><span class="intbl"><em>x<sup>2</sup>−1</em><strong>x−1</strong></span>
|
||
<!-- limx->1 x^2~−1/x−1 -->
|
||
</td>
|
||
<td> <img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"> </td>
|
||
<td class="larger"><span class="intbl"><em>(1−1)</em><strong>(1−1)</strong></span> = <span class="intbl"><em>0</em><strong>0</strong></span></td>
|
||
<td> <img src="../images/style/no.svg" alt="not"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>No luck. Need to try something else.</p>
|
||
</div>
|
||
<p> </p>
|
||
<h3>2. Factors</h3>
|
||
<p>We can try <a href="../algebra/factoring.html">factoring</a>.</p>
|
||
|
||
<div class="example">
|
||
|
||
|
||
|
||
<h3>Example:</h3>
|
||
<p class="center large"><span class="lim"><em>lim</em><strong>x→1</strong></span><span class="intbl"><em>x<sup>2</sup>−1</em><strong>x−1</strong></span></p>
|
||
<p>By factoring <span class="larger">(x<sup>2</sup>−1)</span> into <span class="larger">(x−1)(x+1)</span> we get:</p><br>
|
||
<div class="center large"><span class="lim"><em>lim</em><strong>x→1</strong></span> <span class="intbl"><em>x<sup>2</sup>−1</em><strong>x−1</strong></span> = <span class="lim"><em>lim</em><strong>x→1</strong></span> <span class="intbl"><em>(x−1)(x+1)</em><strong>(x−1)</strong></span>
|
||
</div>
|
||
<!-- LIM[x->1] x^2~-1/x-1 = LIM[x->1] (x-1)(x+1)/(x-1)-->
|
||
<br>
|
||
<div class="center large">= <span class="lim"><em>lim</em><strong>x→1</strong></span> (x+1)</div>
|
||
<!-- = LIM[x->1] (x+1) -->
|
||
<p>Now we can just substitiute x=1 to get the limit:</p>
|
||
|
||
<div class="center large"><span class="lim"><em>lim</em><strong>x→1</strong></span> (x+1) = 1+1 = 2</div>
|
||
<!-- LIM[x->1] (x+1) = 1+1 = 2 -->
|
||
</div>
|
||
<h3>3. Conjugate</h3>
|
||
<p>For some fractions multiplying top and bottom by a <a href="../algebra/conjugate.html">conjugate</a> can help.</p>
|
||
<div class="simple">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td align="right" valign="top">The conjugate is where we change<br>
|
||
the sign in the middle of 2 terms like this:</td>
|
||
<td><img src="../algebra/images/conjugate.svg" alt="conjugate 3x+1 is 3x-1"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>Here is an example where it will help us find a limit:</p>
|
||
</div>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<div class="center larger"><span class="lim"><em>lim</em><strong>x→4</strong></span> <span class="intbl"><em>2−√x</em><strong>4−x</strong></span></div>
|
||
<!-- LIM[x->4] 2-SQRx/4-x --></td>
|
||
<td> </td>
|
||
<td>Evaluating this at x=4 gives 0/0, which is not a good answer!</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>So, let's try some rearranging:</p>
|
||
|
||
|
||
<table align="center" width="91%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td align="right" height="51" width="66%">Multiply top and bottom by the conjugate of the top:</td>
|
||
<td align="right" height="51" width="7%"> </td>
|
||
<td height="51" width="27%">
|
||
<div class="center larger"><span class="intbl"><em>2−√x</em><strong>4−x</strong></span> × <span class="intbl"><em>2+√x</em><strong>2+√x</strong></span></div>
|
||
<!-- 2-SQRx/4-x * 2+SQRx/2+SQRx --></td>
|
||
</tr>
|
||
<tr>
|
||
<td align="right" width="66%"> </td>
|
||
<td align="right" width="7%"> </td>
|
||
<td width="27%"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td align="right" width="66%">Simplify top using <span class="center large">(a+b)(a−b) = a<sup>2</sup> − b<sup>2</sup></span>
|
||
<!-- (a+b)(a-b) = a^2 - b^2 -->:</td>
|
||
<td align="right" width="7%"> </td>
|
||
<td width="27%">
|
||
<div class="center larger"><span class="intbl"><em>2<sup>2</sup> − (√x)<sup>2</sup></em><strong>(4−x)(2+√x)</strong></span></div>
|
||
<!-- 2^2-(SQRx)^2/(4-x)(2+SQRx) --></td>
|
||
</tr>
|
||
<tr>
|
||
<td align="right" width="66%"> </td>
|
||
<td align="right" width="7%"> </td>
|
||
<td width="27%"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td align="right" width="66%">Simplify top further:</td>
|
||
<td align="right" width="7%"> </td>
|
||
<td width="27%">
|
||
<div class="center larger"><span class="intbl"><em>4−x</em><strong>(4−x)(2+√x)</strong></span></div>
|
||
<!-- 4-x/(4-x)(2+SQRx) --></td>
|
||
</tr>
|
||
<tr>
|
||
<td align="right" width="66%"> </td>
|
||
<td align="right" width="7%"> </td>
|
||
<td width="27%"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td align="right" width="66%">Cancel (4−x) from top and bottom: </td>
|
||
<td align="right" width="7%"> </td>
|
||
<td width="27%">
|
||
<div class="center larger"><span class="intbl"><em>1</em><strong>2+√x</strong></span></div>
|
||
<!-- 1/2+SQRx --></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>So, now we have:</p>
|
||
<div class="center larger"><span class="lim"><em>lim</em><strong>x→4</strong></span> <span class="intbl"><em>2−√x</em><strong>4−x</strong></span> = <span class="lim"><em>lim</em><strong>x→4</strong></span> <span class="intbl"><em>1</em><strong>2+√x</strong></span> = <span class="intbl"><em>1</em><strong>2+√4</strong></span> = <span class="intbl"><em>1</em><strong>4</strong></span></div>
|
||
<!-- LIM[x->4] 2-SQRx/4-x = LIM[x->4] 1/2+SQRx = 1/2+SQR4 = 1/4 -->
|
||
<p><b>Done!</b></p>
|
||
<p> </p>
|
||
<h3>4. Infinite Limits and Rational Functions</h3>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;">A <a href="../algebra/rational-expression.html">Rational Function</a> is one that is the ratio of two polynomials:</td>
|
||
<td> </td>
|
||
<td>
|
||
<div class="center larger">f(x) = <span class="intbl"><em>P(x)</em><strong>Q(x)</strong></span></div>
|
||
<!-- f(x) = P(x)/Q(x) --></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"> </td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">For example, here <i><b>P(x) = x<sup>3 </sup>+ 2x − 1</b></i>, and <i><b>Q(x) = 6x<sup>2</sup></b></i>:</td>
|
||
<td> </td>
|
||
<td>
|
||
<div class="center larger"><span class="intbl"><em>x<sup>3</sup> + 2x − 1</em><strong>6x<sup>2</sup></strong></span></div>
|
||
<!-- x^3~+2x-1/6x^2 --></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>By finding the overall <a href="../algebra/degree-expression.html">Degree of the Function</a> we can find out whether the function's limit is 0, Infinity, -Infinity, or easily calculated from the coefficients.</p>
|
||
<p class="center larger">Read more at <a href="limits-infinity.html">Limits To Infinity</a>.</p>
|
||
<p> </p>
|
||
<h3>5. <span class="center">L'Hôpital's Rule</span></h3>
|
||
<p><span class="center">L'Hôpital's Rule</span> can help us evaluate limits that at first seem to be "indeterminate", such as <span class="intbl"><em>0</em><strong>0</strong></span> and <span class="intbl"><em>∞</em><strong>∞</strong></span>.</p>
|
||
<p class="center larger">Read more at <a href="l-hopitals-rule.html">L'Hôpital's Rule</a>.</p>
|
||
<p> </p>
|
||
<h3>6. Formal Method</h3>
|
||
<p>The formal method sets about proving that we can get <b>as close as we want</b> to the answer by making "x" close to "a".</p>
|
||
<p class="center larger">Read more at <a href="limits-formal.html">Limits (Formal Definition)</a></p>
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script>
|
||
getQ(6770, 6771, 6772, 6773, 6774, 6775, 6776, 6777, 6778, 6779);
|
||
</script> </div>
|
||
|
||
<div class="related">
|
||
<a href="limits-infinity.html">Limits To Infinity</a>
|
||
<a href="index.html">Calculus Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2020 MathsIsFun.com</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/calculus/limits-evaluating.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:48:57 GMT -->
|
||
</html> |