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412 lines
14 KiB
HTML
412 lines
14 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/calculus/limits.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:48:55 GMT -->
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<head>
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<!-- #BeginEditable "doctitle" -->
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<title>Limits (An Introduction)</title>
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<script src="images/limit-runner.js" type="text/javascript"></script>
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<style>
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text-align: center;
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margin: 0 4px 0 2px;
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border-collapse: collapse;
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}
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.lim em {
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display: table-row;
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text-align: center;
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font-style: inherit;
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}
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.lim strong {
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display: table-row;
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text-align: center;
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font-weight: inherit;
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font-size: 80%;
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line-height: 9px;
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}
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</style>
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<meta name="keywords" content="math, maths, mathematics, school, homework, education">
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<script src="../main4.js"></script>
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<script>document.write(gTagHTML())</script>
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<body id="bodybg" class="adv">
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<div id="stt"></div>
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<div id="adTop"></div>
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<header>
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<div id="hdr"></div>
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<div id="tran"></div>
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</header>
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<nav>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Limits <i>(An Introduction)</i></h1>
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<h2>Approaching ...</h2> Sometimes we can't work something out directly ... but we <b>can</b> see what it should be as we get closer and closer!
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<div class="example">
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<h3>Example:</h3>
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<p class="center larger"><span class="intbl">
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<em>(x<sup>2</sup> − 1)</em>
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<strong>(x − 1)</strong>
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</span></p>
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<p>Let's work it out for x=1:</p>
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<p class="center larger"><span class="intbl">
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<em>(1<sup>2 </sup>− 1)</em>
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<strong>(1 − 1)</strong>
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</span> = <span class="intbl">
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<em>(1 − 1)</em>
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<strong>(1 − 1)</strong>
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</span> = <span class="intbl">
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<em>0</em>
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<strong>0</strong>
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</span></p>
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</div>
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<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>
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<p>So instead of trying to work it out for x=1 let's try <b>approaching</b> it closer and closer:</p>
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<div class="example">
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<h3>Example Continued:</h3>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:right;">
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<td class="larger">x</td>
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<td style="width:30px;"> </td>
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<td class="larger"><span class="intbl">
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<em>(x<sup>2</sup> − 1)</em>
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<strong>(x − 1)</strong>
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</span></td>
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</tr>
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<tr style="text-align:right;">
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<td>0.5</td>
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<td> </td>
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<td>1.50000</td>
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</tr>
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<tr style="text-align:right;">
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<td>0.9</td>
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<td> </td>
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<td>1.90000</td>
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</tr>
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<tr style="text-align:right;">
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<td>0.99</td>
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<td> </td>
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<td>1.99000</td>
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</tr>
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<tr style="text-align:right;">
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<td>0.999</td>
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<td> </td>
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<td>1.99900</td>
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</tr>
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<tr style="text-align:right;">
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<td>0.9999</td>
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<td> </td>
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<td>1.99990</td>
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</tr>
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<tr style="text-align:right;">
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<td>0.99999</td>
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<td> </td>
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<td>1.99999</td>
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</tr>
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<tr style="text-align:right;">
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<td>...</td>
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<td> </td>
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<td>...</td>
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</tr>
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</tbody></table>
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</div>
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<p>Now we see that as x gets close to 1, then <span class="intbl">
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<em>(x<sup>2</sup>−1)</em>
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<strong>(x−1)</strong>
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</span> gets <b>close to 2</b></p>
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</div>
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<p>We are now faced with an interesting situation:</p>
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<ul>
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<li>When x=1 we don't know the answer (it is <b>indeterminate</b>)</li>
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<li>But we can see that it is <b>going to be 2</b></li>
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</ul>
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<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>
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<p class="center larger">The <b>limit</b> of <span class="intbl">
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<em>(x<sup>2</sup>−1)</em>
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<strong>(x−1)</strong>
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</span> as x approaches 1 is<b> 2</b></p>
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<p>And it is written in symbols as:</p>
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<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>
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<!-- limx->1 x^2~−1/x−1 = 2 -->
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<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>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">
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<p>As a graph it looks like this:</p>
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<p>So, in truth, we <b>cannot say what the value at x=1 is.</b></p>
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<p>But we <b>can</b> say that as we approach 1, <b>the limit is 2.</b></p>
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</td>
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<td style="text-align:right;"> </td>
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<td><img src="images/graph-x2-1-x-1.svg" alt="graph hole"></td>
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</tr>
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</tbody></table>
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<h2>Test Both Sides!</h2>
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<div style="float:left; margin: 0 10px 5px 0;">
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<script>
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limitrunnerMain();
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</script>
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</div>
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<p>It is like running up a hill and then finding the path<b> is magically "not there"...</b></p>
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<p>... but if we only check one side, who knows what happens?</p>
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<p>So we need to test it <b>from both directions</b> to be sure where it "should be"!</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example Continued</h3>
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<p>So, let's try from the other side:</p>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:right;">
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<td class="larger">x</td>
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<td> </td>
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<td class="larger"><span class="intbl">
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<em>(x<sup>2</sup> − 1)</em>
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<strong>(x − 1)</strong>
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</span></td>
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</tr>
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<tr style="text-align:right;">
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<td>1.5</td>
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<td> </td>
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<td>2.50000</td>
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</tr>
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<tr style="text-align:right;">
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<td>1.1</td>
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<td> </td>
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<td>2.10000</td>
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</tr>
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<tr style="text-align:right;">
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<td>1.01</td>
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<td> </td>
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<td>2.01000</td>
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</tr>
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<tr style="text-align:right;">
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<td>1.001</td>
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<td> </td>
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<td>2.00100</td>
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</tr>
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<tr style="text-align:right;">
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<td>1.0001</td>
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<td> </td>
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<td>2.00010</td>
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</tr>
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<tr style="text-align:right;">
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<td>1.00001</td>
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<td> </td>
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<td>2.00001</td>
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</tr>
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<tr style="text-align:right;">
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<td>...</td>
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<td> </td>
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<td>...</td>
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</tr>
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</tbody></table>
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</div>
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<p>Also heading for 2, so that's OK</p>
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</div>
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<h2>When it is different from different sides</h2>
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<p style="float:right; margin: 0 0 5px 10px; border: 1px solid blue;"><img src="images/discontinuous-function.svg" alt="discontinuous function"></p>
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<p>How about a function <b>f(x)</b> with a "break" in it like this:</p>
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<p class="center large">The limit does not exist at "a"</p>
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<p><b>We can't say what the value at "a" is</b>, because there are two competing answers:</p>
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<ul>
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<li>3.8 from the left, and</li>
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<li>1.3 from the right</li>
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</ul>
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<p>But we <b>can</b> use the special "−" or "+" signs (as shown) to define one sided limits:</p>
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<ul>
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<li>the <b>left-hand</b> limit (−) is 3.8</li>
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<li>the <b>right-hand</b> limit (+) is 1.3</li>
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</ul>
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<p>And the ordinary limit <b>"does not exist"</b></p>
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<h2>Are limits only for difficult functions?</h2>
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<p>Limits can be used even when we <b>know the value when we get there</b>! Nobody said they are only for difficult functions.</p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center large"><span class="lim"><em>lim</em><strong>x→10</strong></span><span class="intbl"><em>x</em><strong>2</strong></span> = 5</p>
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<!-- limx->10 x/2 = 5 -->
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<p>We know perfectly well that 10/2 = 5, but limits can still be used (if we want!)</p>
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</div>
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<h2>Approaching Infinity</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../sets/images/infinity.svg" alt="infinity"></p>
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<p><a href="../numbers/infinity.html">Infinity</a> is a very special idea.
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We know we can't reach it, but we can still try to work out the value of
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functions that have infinity in them.</p>
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<h3>Let's start with an interesting example.</h3>
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<div class="simple">
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<table align="center" width="400" border="0">
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<tbody>
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<tr>
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<td class="larger">Question: What is the value of <span class="intbl"><em>1</em><strong><span class="times">∞</span></strong>
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</span> ?</td>
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</tr>
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</tbody></table><br>
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<table align="center" width="400" border="0">
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<tbody>
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<tr>
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<td class="large">Answer: We don't know!</td>
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</tr>
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</tbody></table>
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</div>
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<p class="center large"> </p>
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<h3>Why Don't We Know?</h3>
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<p>The simplest reason is that Infinity is not a number, it is an idea.</p>
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<p>So <span class="intbl"><em>1</em><strong><span class="times">∞</span></strong>
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</span> is a bit like saying <span class="intbl">
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<em>1</em>
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<strong>beauty</strong>
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</span> or <span class="intbl">
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<em>1</em>
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<strong>tall</strong>
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</span>.</p>
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<p>Maybe we could say that <span class="intbl"><em>1</em><strong><span class="times">∞</span></strong>
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</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>
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<p class="center">In fact <span class="intbl"><em>1</em><strong><span class="times">∞</span></strong>
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</span> is known to be <b>undefined</b>.</p>
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<h3>But We Can Approach It!</h3>
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<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>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../sets/images/function-reciprocal-pos.svg" alt="graph 1/x"></p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:right;">
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<td style="width:150px;"><b>x</b></td>
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<td style="width:150px;"><b><span class="intbl">
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<em>1</em>
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<strong>x</strong>
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</span></b></td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:150px;">1</td>
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<td style="width:150px;">1.00000</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:150px;">2</td>
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<td style="width:150px;">0.50000</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:150px;">4</td>
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<td style="width:150px;">0.25000</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:150px;">10</td>
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<td style="width:150px;">0.10000</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:150px;">100</td>
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<td style="width:150px;">0.01000</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:150px;">1,000</td>
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<td style="width:150px;">0.00100</td>
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</tr>
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<tr style="text-align:right;">
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<td style="width:150px;">10,000</td>
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<td style="width:150px;">0.00010</td>
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</tr>
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</tbody></table>
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<p>Now we can see that as x gets larger, <b><span class="intbl">
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<em>1</em>
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<strong>x</strong>
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</span></b> tends towards 0</p>
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<p>We are now faced with an interesting situation:</p>
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<ul>
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<li>We can't say what happens when x gets to infinity</li>
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<li>But we can see that <b><span class="intbl">
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<em>1</em>
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||
<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>
|
||
<p class="center large"><span class="lim"><em>lim</em><strong>x→∞</strong></span><span class="intbl"><em>1</em><strong>x</strong></span> = 0</p>
|
||
<!-- limx->INF 1/x = 0 -->
|
||
<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 class="center large"> </p>
|
||
<div class="def">
|
||
<p class="center large"><i>When you see "limit", think "approaching"</i></p>
|
||
</div>
|
||
<p class="center large"> </p>
|
||
<p>It is a mathematical way of saying <i>"we are not talking about when x=<span class="times">∞</span>, but we know as x gets bigger, the answer gets closer and closer to <b>0</b>"</i>.</p>
|
||
<p class="center">Read more at <a href="limits-infinity.html">Limits to Infinity</a>.</p>
|
||
<h2>Solving!</h2>
|
||
<p>We have been a little lazy so far, and just said that a limit equals some value because it <b>looked like it was going to</b>.</p>
|
||
<p>That is not really good enough! Read more at <a href="limits-evaluating.html">Evaluating Limits</a>.</p>
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script>
|
||
getQ(6760, 6761, 6762, 6763, 6764, 6765, 6766, 6767, 6768, 6769);
|
||
</script> </div>
|
||
|
||
<div class="related">
|
||
<a href="limits-evaluating.html">Evaluating Limits</a>
|
||
<a href="index.html">Calculus Index</a>
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