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381 lines
16 KiB
HTML
381 lines
16 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
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<title>Limits (Formal Definition)</title>
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<meta name="description" content="Sometimes we cant work something out directly ... but we can see what it should be as we get closer and closer">
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.del, .eps {padding:2px; vertical-align:baseline; }
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Limits <i>(Formal Definition)</i></h1>
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<div class="center"><i>Please read <a href="limits.html">Introduction to Limits</a> first</i></div>
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<h2>Approaching ...</h2>
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<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>
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<div class="example">
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<h3>Example:</h3>
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<p class="center large"><span class="intbl"> <em>(x<sup>2</sup> − 1)</em> <strong>(x − 1)</strong> </span></p>
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<p>Let's work it out for x=1:</p>
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<p class="center large"><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>
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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="large">x</td>
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<td style="width:30px;"> </td>
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<td class="large"><span class="intbl"> <em>(x<sup>2</sup> − 1)</em> <strong>(x − 1)</strong> </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"> <em>(x<sup>2</sup>−1)</em> <strong>(x−1)</strong> </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 large">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>
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<p>And it is written in symbols as:</p>
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<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> = 2</div>
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<!-- LIM[x-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></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" height="147" width="137"></td>
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</tr>
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</tbody></table>
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<h2>More Formal</h2>
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<p>But instead of saying a limit equals some value because it <b>looked like it was going to</b>, we can have a more formal definition.</p>
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<p>So let's start with the general idea.</p>
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<h2>From English to Mathematics</h2>
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<p>Let's say it in English first:</p>
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<p class="center large">"f(x) gets close to <i>some limit</i> as x gets close to some value"</p>
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<p>When we call the Limit "L", and the value that x gets close to "a" we can say</p>
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<p class="center large">"f(x) gets close to L as x gets close to a"</p>
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<p class="center large"><img src="images/limit-idea.svg" alt="limit idea: f(x) goes to L as x goes to a" height="27" width="314"></p>
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<h2>Calculating "Close"</h2>
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<p>Now, what is a mathematical way of saying "close" ... could we subtract one value from the other?</p>
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<div class="example">
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<p>Example 1: 4.01 − 4 = 0.01 (that looks good)<br>
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Example 2: 3.8 − 4 = −0.2 (<b>negatively</b> close?)</p>
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</div>
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<p>So how do we deal with the negatives? We don't care about positive or negative, we just want to know how far ... which is the <a href="../number-line.html#absolute">absolute value</a>.</p>
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<p class="center large">"How Close" = |a−b|</p>
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<div class="example">
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<p>Example 1: |4.01−4| = 0.01 <img src="../images/style/yes.svg" alt="yes" height="30" width="30"><br>
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Example 2: |3.8−4| = 0.2 <img src="../images/style/yes.svg" alt="yes" height="30" width="30"></p>
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</div>
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<p>And when |a−b| is small we know we are close, so we write:</p>
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<p class="center large">"|f(x)−L| is small when |x−a| is small"</p>
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<p>And this animation shows what happens with the function</p>
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<p class="center large">f(x) = <span class="intbl"> <em>(x<sup>2</sup>−1)</em> <strong>(x−1)</strong> </span></p>
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<div class="script" style="height: 400px;">
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images/limit-lines.js
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</div>
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<p class="center">f(x) approaches L=2 as x approaches a=1,<br>
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so |f(x)−2| is small when |x−1| is small.</p>
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<h2>Delta and Epsilon</h2>
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<p>But "small" is still English and not "Mathematical-ish".</p>
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<p>Let's choose two values <b>to be smaller than</b>:</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><span class="del">δ</span></td>
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<td style="width:20px;"> </td>
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<td>that |x−a| must be smaller than</td>
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</tr>
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<tr>
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<td><span class="eps">ε</span></td>
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<td> </td>
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<td> that |f(x)−L| must be smaller than</td>
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</tr>
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</tbody></table>
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<p class="center"><i>Note: those two Greek letters (δ is <i>"delta"</i> and ε is <i>"epsilon")</i> are<br>
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so often used we get the phrase "<b>delta-epsilon</b>"</i></p>
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<p>And we have:</p>
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<p class="center large">|f(x)−L|<<span class="eps">ε</span> when |x−a|<<span class="del">δ</span></p>
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<p><b>That actually says it!</b> So if you understand that you understand limits ...</p>
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<p>... but to be <b>absolutely precise</b> we need to add these conditions:</p>
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<ul>
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<li>it is true for any <span class="eps">ε</span>>0</li>
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<li><span class="del">δ</span> exists, and is >0</li>
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<li>x is <b>not equal to</b> a, meaning 0<|x−a|</li>
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</ul>
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<p>And this is what we get:</p>
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<div class="def">
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<p class="center large">For any <span class="eps">ε</span>>0, there is a <span class="del">δ</span>>0 so that |f(x)−L|<<span class="eps">ε</span> when 0<|x−a|<<span class="del">δ</span></p>
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</div>
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<p>That is the formal definition. It actually looks pretty scary, doesn't it?</p>
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<p>But in essence it says something simple:</p>
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<p class="center"><i><b>f(x) gets close to L</b></i> when <b><i>x gets close to a</i></b></p>
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<h2>How to Use it in a Proof</h2>
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<p>To use this definition in a proof, we want to go</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td style="width:150px;">From:</td>
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<td> </td>
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<td style="width:150px;">To:</td>
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</tr>
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<tr>
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<td class="large" align="center" width="150">0<|x−a|<<span class="del">δ</span></td>
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<td class="large" align="center"><img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></td>
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<td class="large" align="center" width="150">|f(x)−L|<<span class="eps">ε</span></td>
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</tr>
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</tbody></table>
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<p>This usually means finding a formula for <span class="del">δ</span> (in terms of <span class="eps">ε</span>) that works.</p>
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<p>How do we find such a formula?</p>
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<p class="center"><span class="large">Guess and Test!</span></p>
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<p>That's right, we can:</p>
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<ol>
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<li>Play around till we find a formula that <b>might</b> work</li>
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<li><b>Test</b> to see if that formula does work</li>
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</ol>
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<h2>Example: Let's try to show that</h2>
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<div class="center large"><span class="lim"><em>lim</em><strong>x→3</strong></span> 2x+4 = 10</div>
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<p>Using the letters we talked about above:</p>
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<ul>
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<li>The value that x approaches, "a", is 3</li>
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<li>The Limit "L" is 10</li>
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</ul>
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<p>So we want to know how we go from:</p>
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<p class="center large">0<|x−3|<<span class="del">δ </span><br>
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to<br>
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|(2x+4)−10|<<span class="eps">ε</span></p>
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<h3>Step 1: Play around till you find a formula that <b>might</b> work</h3>
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<div class="tbl">
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<div class="row">
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<span class="lt">Start with:</span>
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<span class="rt">|(2x+4)−10| < <span class="eps">ε</span></span>
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</div>
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<div class="row">
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<span class="lt">Simplify:</span>
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<span class="rt">|2x−6| < <span class="eps">ε</span></span>
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</div>
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<div class="row">
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<span class="lt">Move 2 outside ||:</span>
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<span class="rt">2|x−3| < <span class="eps">ε</span></span>
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</div>
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<div class="row">
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<span class="lt">Divide both sides by 2:</span>
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<span class="rt">|x−3| < <span class="eps">ε</span>/2</span>
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</div>
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</div>
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<p>So we can now guess that <b><span class="del">δ</span>=<span class="eps">ε</span>/2</b> might work</p>
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<h3>Step 2: <b>Test</b> to see if that formula works.</h3>
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<p>So, can we get from <b>0<|x−3|<<span class="del">δ</span></b> to <b>|(2x+4)−10|<<span class="eps">ε</span></b> ... ?</p>
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<p>Let's see ...</p>
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<div class="tbl">
|
||
<div class="row">
|
||
<span class="lt">Start with:</span>
|
||
<span class="rt">0 < |x−3| < <span class="del">δ</span></span>
|
||
</div>
|
||
<div class="row">
|
||
<span class="lt">Replace <span class="del">δ</span> with <span class="eps">ε</span>/2:</span>
|
||
<span class="rt">0 < |x−3| < <span class="eps">ε</span>/2</span>
|
||
</div>
|
||
<div class="row">
|
||
<span class="lt">Multiply all by 2:</span>
|
||
<span class="rt">0 < 2|x−3| < <span class="eps">ε</span></span>
|
||
</div>
|
||
<div class="row">
|
||
<span class="lt">Move 2 inside the ||:</span>
|
||
<span class="rt">0 < |2x−6| < <span class="eps">ε</span></span>
|
||
</div>
|
||
<div class="row">
|
||
<span class="lt">Replace "−6" with "+4−10":</span>
|
||
<span class="rt">0 < |(2x+4)−10| < <span class="eps">ε</span></span>
|
||
</div>
|
||
</div>
|
||
<p>Yes! We can go from <b>0<|x−3|<<span class="del">δ</span></b> to <b>|(2x+4)−10|<<span class="eps">ε</span></b> by choosing <span class="del">δ</span>=<span class="eps">ε</span>/2</p>
|
||
<p class="center large">DONE!</p>
|
||
<p>We have seen then that given <span class="eps">ε</span> we can find a <span class="del">δ</span>, so it is true that:</p>
|
||
<p class="center">For any<span class="eps"> ε</span>, there is a <span class="del">δ</span> so that |f(x)−L|<<span class="eps">ε</span> when 0<|x−a|<<span class="del">δ</span></p>
|
||
<p>And we have proved that</p>
|
||
<div class="center large"><span class="lim"><em>lim</em><strong>x→3</strong></span> 2x+4 = 10</div>
|
||
<!-- LIM[x-3] 2x+4 = 10 -->
|
||
|
||
|
||
<h2>Conclusion</h2>
|
||
|
||
<p>That was a fairly simple proof, but it hopefully explains the strange "there is a ..." wording, and it does show a good way of approaching these kind of proofs.</p>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="limits.html">Introduction to Limits</a>
|
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<a href="index.html">Calculus Index</a>
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