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
472 lines
20 KiB
HTML
472 lines
20 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/calculus/limits-infinity.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:48:56 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Limits to Infinity</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" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</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 to Infinity</h1>
|
||
|
||
<p class="center"><i>Please read <a href="limits.html">Limits (An Introduction)</a> first</i></p>
|
||
|
||
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="../sets/images/infinity.svg" alt="infinity"></p>
|
||
<p><br>
|
||
<a href="../numbers/infinity.html">Infinity</a> is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them.</p>
|
||
|
||
|
||
<h2>One Divided By Infinity</h2>
|
||
<p>Let's start with an interesting example.</p>
|
||
<div class="simple">
|
||
<table align="center" width="400" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td class="larger">Question: What is the value of <span class="intbl">
|
||
<em>1</em>
|
||
<strong><span class="times">∞</span></strong>
|
||
</span> ?</td>
|
||
</tr>
|
||
</tbody></table><br>
|
||
<table align="center" width="400" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td class="large">Answer: We don't know!</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p class="center large"> </p>
|
||
<h3>Why don't we know?</h3>
|
||
<p>The simplest reason is that Infinity is not a number, it is an idea.</p>
|
||
<p>So <span class="intbl">
|
||
<em>1</em>
|
||
<strong><span class="times">∞</span></strong>
|
||
</span> is a bit like saying <span class="intbl">
|
||
<em>1</em>
|
||
<strong>beauty</strong>
|
||
</span> or <span class="intbl">
|
||
<em>1</em>
|
||
<strong>tall</strong>
|
||
</span>.</p>
|
||
<p>Maybe we could say that <span class="intbl">
|
||
<em>1</em>
|
||
<strong><span class="times">∞</span></strong>
|
||
</span>= 0, ... but that is a problem too, because if we divide 1 into infinite pieces and they end up 0 each, what happened to the 1?</p>
|
||
<p class="center">In fact <span class="intbl">
|
||
<em>1</em>
|
||
<strong><span class="times">∞</span></strong>
|
||
</span> is known to be <b>undefined</b>.</p>
|
||
<h3>But We Can Approach It!</h3>
|
||
<p>So instead of trying to work it out for infinity (because we can't get a sensible answer), let's try larger and larger values of x:</p>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="../sets/images/function-reciprocal-pos.svg" alt="graph 1/x"></p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:right;">
|
||
<td><b>x</b></td>
|
||
<td><b><span class="intbl">
|
||
<em>1</em>
|
||
<strong>x</strong>
|
||
</span></b></td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:150px;">1</td>
|
||
<td style="width:150px;">1.00000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:150px;">2</td>
|
||
<td style="width:150px;">0.50000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:150px;">4</td>
|
||
<td style="width:150px;">0.25000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:150px;">10</td>
|
||
<td style="width:150px;">0.10000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:150px;">100</td>
|
||
<td style="width:150px;">0.01000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:150px;">1,000</td>
|
||
<td style="width:150px;">0.00100</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:150px;">10,000</td>
|
||
<td style="width:150px;">0.00010</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>Now we can see that as x gets larger, <b><span class="intbl">
|
||
<em>1</em>
|
||
<strong>x</strong>
|
||
</span></b> tends towards 0</p>
|
||
<p>We are now faced with an interesting situation:</p>
|
||
<ul>
|
||
<li>We can't say what happens when x gets to infinity</li>
|
||
<li>But we can see that <b><span class="intbl">
|
||
<em>1</em>
|
||
<strong>x</strong>
|
||
</span></b> is <b>going towards 0</b></li>
|
||
</ul>
|
||
<p>We want to give the answer "0" 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 <b><span class="intbl">
|
||
<em>1</em>
|
||
<strong>x</strong>
|
||
</span></b> as x approaches Infinity is<b> 0</b></p>
|
||
<p>And write it like this:</p>
|
||
<div class="center larger"><span class="lim"><em>lim</em><strong>x→∞</strong></span> (<span class="intbl"><em>1</em><strong>x</strong></span>) = 0</div>
|
||
<p><br></p>
|
||
<p>In other words:</p>
|
||
<p class="center large">As x approaches infinity, then <b><span class="intbl">
|
||
<em>1</em>
|
||
<strong>x</strong>
|
||
</span></b> approaches 0</p>
|
||
<p><i>When you see "limit", think "approaching"</i></p>
|
||
<p class="center large"> </p>
|
||
<p>It is a mathematical way of saying <i>"we are not talking about when x=<span style="font-size: 20px; font-family: serif;">∞</span>, but we know as x gets bigger, the answer gets closer and closer to <b>0</b>"</i>.</p>
|
||
<h3>Summary</h3>
|
||
<p>So, sometimes Infinity cannot be used directly, but we <b>can</b> use a limit.</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>What happens <b>at</b> <span style="font-size: 20px; font-family: serif;">∞</span> is <b>undefined</b> ...</td>
|
||
<td> </td>
|
||
<td style="text-align: center; "><b><span class="intbl">
|
||
<em>1</em>
|
||
<strong>∞</strong>
|
||
</span></b></td>
|
||
<td><br>
|
||
</td>
|
||
<td style="text-align: center; "><img src="../images/style/no.svg" alt="not"></td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td> </td>
|
||
<td style="text-align: center;"> </td>
|
||
<td> </td>
|
||
<td style="text-align: center;"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td>... but we do know that <b>1/x approaches 0</b><br>
|
||
as <b>x approaches infinity</b></td>
|
||
<td> </td>
|
||
<td style="text-align: center;">
|
||
<div class="center larger"><span class="lim"><em>lim</em><strong>x→∞</strong></span> (<span class="intbl"><em>1</em><strong>x</strong></span>) = 0</div>
|
||
</td>
|
||
<td> </td>
|
||
<td style="text-align: center;"><img src="../images/style/yes.svg" alt="yes"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
<h2>Limits Approaching Infinity</h2>
|
||
<p>What is the limit of this function as x approaches infinity?</p>
|
||
<p class="center large">y = 2x</p>
|
||
<p>Obviously as "x" gets larger, so does "2x":</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;"><b>x</b></td>
|
||
<td style="width:100px;"><b>y=2x</b></td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">1</td>
|
||
<td style="width:100px;">2</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">2</td>
|
||
<td style="width:100px;">4</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">4</td>
|
||
<td style="width:100px;">8</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">10</td>
|
||
<td style="width:100px;">20</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">100</td>
|
||
<td style="width:100px;">200</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">...</td>
|
||
<td style="width:100px;">...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>So as "x" approaches infinity, then "2x" also approaches infinity. We write this:</p>
|
||
<div class="center larger"><span class="lim"><em>lim</em><strong>x→∞</strong></span> 2x = ∞</div>
|
||
<p class="center80"><img src="../images/style/info.svg" style="float:left; margin: 0 10px 5px 0;" alt="info">
|
||
But don't be fooled by the "=". We cannot actually <b>get</b> to infinity, but in "limit" language the <b>limit is infinity</b> (which is really saying the function is limitless).</p>
|
||
|
||
|
||
|
||
<h2>Infinity and Degree</h2>
|
||
<p>We have seen two examples, one went to 0, the other went to infinity.</p>
|
||
<p>In fact many infinite limits are actually quite easy to work out, when we figure out "which way it is going", like this:</p>
|
||
<p class="center80"><img src="../images/style/zero.svg" style="float:left; margin: 0 10px 5px 0;" alt="zero" height="46" width="46">Functions like <b>1/x</b> approach <b>0</b> as x approaches infinity. This is also true for 1/x<sup>2</sup> etc</p>
|
||
<p class="center80"><img src="../images/style/up.svg" style="float:left; margin: 0 10px 5px 0;" alt="up" height="46" width="46">A function such as <b>x</b> will approach infinity, as well as <b>2x</b>, or <b>x/9</b> and so on. Likewise functions with <b>x<sup>2</sup></b> or <b>x<sup>3</sup></b> etc will also approach infinity.</p>
|
||
<p class="center80"><img src="../images/style/down.svg" style="float:left; margin: 0 10px 5px 0;" alt="down" height="46" width="46">But be careful, a function like "<b>−x</b>" will approach "<b>−infinity</b>", so we have to look at the signs of <b>x</b>.</p><br>
|
||
<div class="example">
|
||
<h3>Example: <b>2x<sup>2</sup>−5x</b></h3>
|
||
<ul>
|
||
<li><b>2x<sup>2</sup></b> will head towards +infinity</li>
|
||
<li><b>−5x</b> will head towards -infinity</li>
|
||
<li>But <b>x<sup>2</sup></b> grows more rapidly than <b>x</b>, so <b>2x<sup>2</sup>−5x</b> will head towards +infinity</li>
|
||
</ul>
|
||
</div>
|
||
<p>In fact, when we look at the <a href="../algebra/degree-expression.html">Degree</a> of the function (the highest <a href="../exponent.html">exponent</a> in the function) we can tell what is going to happen:</p>
|
||
<p>When the Degree of the function is:</p>
|
||
<ul>
|
||
<li>greater than 0, the limit is <b>infinity</b> (or <b>−infinity</b>)</li>
|
||
<li>less than 0, the limit is <b>0</b></li>
|
||
</ul>
|
||
<p>But if the <b>Degree is 0 or unknown</b> then we need to work a bit harder to find a limit.</p>
|
||
<h2>Rational Functions</h2>
|
||
<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 style="width:20px;"> </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>Following on from our idea of the <a href="../algebra/degree-expression.html">Degree of the Equation</a>, the first step to find the limit is to ...</p>
|
||
<h2>Compare the <b>Degree of P(x)</b> to the <b>Degree of Q(x)</b>:</h2>
|
||
<div class="dotpoint">If the Degree of P is less than the Degree of Q ...</div>
|
||
|
||
<p class="center large">... the limit is 0.</p>
|
||
|
||
|
||
<div class="dotpoint">If the Degree of P and Q are <b>the same</b> ...</div>
|
||
<p class="center large">... divide the coefficients of the <i>terms with the largest exponent</i>, like this:</p>
|
||
|
||
|
||
|
||
<div class="script">../algebra/images/degree-example.js?mode=lim53</div>
|
||
<div class="script">../algebra/images/degree-example.js?mode=lim4m2</div>
|
||
|
||
|
||
<p class="center">(note that the largest exponents are equal, as the degree is equal)</p>
|
||
<div class="dotpoint">If the Degree of P is greater than the Degree of Q ...</div>
|
||
<p class="center80"><img src="../images/style/up.svg" style="float:left; margin: 0 10px 5px 0;" alt="up" height="46" width="46">... then the limit is positive infinity ...</p>
|
||
<p class="center80"><img src="../images/style/down.svg" style="float:left; margin: 0 10px 5px 0;" alt="down" height="46" width="46">... or maybe negative infinity. <b>We need to look at the signs!</b></p>
|
||
|
||
|
||
<p>We can work out the sign (positive or negative) by looking at the signs of the <i>terms with the largest exponent</i>, just like how we found the coefficients above:</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<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>
|
||
<td> </td>
|
||
<td>
|
||
<p>For example this will go to positive infinity, because both ...</p>
|
||
<ul>
|
||
<li>x<sup>3</sup> <i>(the term with the largest exponent in the top)</i> and</li>
|
||
<li>6x<sup>2</sup> <i>(the term with the largest exponent in the bottom)</i></li>
|
||
</ul> ... are positive.</td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<div class="center larger"><span class="intbl"><em>−2x<sup>2</sup> + x</em><strong>5x − 3</strong></span></div>
|
||
<!-- −2x^2~+x/5x−3 -->
|
||
</td>
|
||
<td> </td>
|
||
<td>But this will head for negative infinity, because −2/5 is negative.</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<h2>A Harder Example: Working Out "e"</h2>
|
||
<p>This formula gets <b>closer</b> to the value of <a href="../numbers/e-eulers-number.html">e (Euler's number)</a> as <b>n increases</b>:</p>
|
||
<div class="center larger">(1 + <span class="intbl"><em>1</em><strong>n</strong></span>)<sup>n</sup></div>
|
||
<!---− (1 + 1/n )^n ---->
|
||
<p>At infinity:</p>
|
||
<div class="center larger">(1 + <span class="intbl"><em>1</em><strong>∞</strong></span> )<sup>∞</sup> = ???</div>
|
||
<!---− (1 + 1/INF )^INF = ??? ---->
|
||
<p class="center large">We don't know!</p>
|
||
|
||
<p>So instead of trying to work it out for infinity (because we can't get a sensible answer), let's try larger and larger values of n:</p>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/graph-1-1-n-n.gif" alt="graph of (1+1/n)^n tends to e" height="218" width="357"></p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:right;">
|
||
<th width="100"><span class="large">n</span></th>
|
||
<th width="150"><span class="large">(1 + 1/n)<sup>n</sup></span></th>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">1</td>
|
||
<td style="width:150px;">2.00000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">2</td>
|
||
<td style="width:150px;">2.25000</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">5</td>
|
||
<td style="width:150px;">2.48832</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">10</td>
|
||
<td style="width:150px;">2.59374</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">100</td>
|
||
<td style="width:150px;">2.70481</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">1,000</td>
|
||
<td style="width:150px;">2.71692</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">10,000</td>
|
||
<td style="width:150px;">2.71815</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<td style="width:100px;">100,000</td>
|
||
<td style="width:150px;">2.71827</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div style="clear:both"></div>
|
||
<p>Yes, it is heading towards the value <b>2.71828...</b> which is <a href="../numbers/e-eulers-number.html">e (Euler's Number)</a></p>
|
||
<p>So again we have an odd situation:</p>
|
||
<ul>
|
||
<li>We don't know what the value is when n=infinity</li>
|
||
<li>But we can see that it settles towards 2.71828...</li>
|
||
</ul>
|
||
<p>So we use limits to write the answer like this:</p>
|
||
<div class="center larger"><span class="lim"><em>lim</em><strong>n→∞</strong></span> (1 + <span class="intbl"><em>1</em><strong>n</strong></span>)<sup>n</sup> = <i><b>e</b></i></div>
|
||
<!-- LIM[n-INF] (1 + 1/n )^n = e -->
|
||
|
||
<p>It is a mathematical way of saying <i>"we are not talking about when n=<span style="font-size: 20px; font-family: serif;">∞</span>, but we know as n gets bigger, the answer gets closer and closer to the value of <b>e</b>"</i>.</p>
|
||
<div class="center80">
|
||
|
||
<h3>Don't Do It The Wrong Way ... !</h3>
|
||
|
||
<p>If we try to use infinity as a "very large real number" (<i><b>it isn't!</b></i>) we get:</p>
|
||
<div class="center large">(1 + <span class="intbl"><em>1</em><strong>∞</strong></span>)<sup>∞</sup> = (1+0)<sup>∞</sup> = 1<sup>∞</sup> = 1
|
||
<img src="../images/style/no.svg" alt="not"> (Wrong!)
|
||
|
||
</div>
|
||
<!---− (1 + 1/INF )^INF = (1+0)^INF = (1)^INF = 1 ---->
|
||
|
||
<p>So don't try using Infinity as a real number: you can get <b>wrong answers</b>!</p>
|
||
<p>Limits are the right way to go.</p>
|
||
|
||
</div>
|
||
<h2>Evaluating Limits</h2>
|
||
<p>I have taken a gentle approach to limits so far, and shown tables and graphs to illustrate the points.</p>
|
||
<p>But to "evaluate" (in other words calculate) the value of a limit can take a bit more effort. Find out more at <a href="limits-evaluating.html">Evaluating Limits</a>.</p>
|
||
<p> </p>
|
||
<div class="questions">6780, 6781, 6782, 6783, 6784, 6785, 6786, 6787, 6788, 6789</div>
|
||
|
||
<div class="related">
|
||
<a href="limits.html">Limits (An Introduction)</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 © 2021 MathsIsFun.com</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/calculus/limits-infinity.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:48:57 GMT -->
|
||
</html> |