http://mlwiki.org/index.php?title=L%27Hopital%27s_Rule&feed=atom&action=history
L'Hopital's Rule - Revision history
2024-03-29T05:00:17Z
Revision history for this page on the wiki
MediaWiki 1.25.3
http://mlwiki.org/index.php?title=L%27Hopital%27s_Rule&diff=749&oldid=prev
Alexey: /* Other Cases */
2016-01-02T11:05:42Z
<p><span dir="auto"><span class="autocomment">Other Cases</span></span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 11:05, 2 January 2016</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L43" >Line 43:</td>
<td colspan="2" class="diff-lineno">Line 43:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>=== Other Cases ===</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>=== Other Cases ===</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>See http://calculus.seas.upenn.edu/?n=Main.LHopitalsRule</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">* </ins>See http://calculus.seas.upenn.edu/?n=Main.LHopitalsRule</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Links ==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* http://math.stackexchange.com/questions/584889/the-intuition-behind-lhopitals-rule/</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Sources ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Sources ==</div></td></tr>
</table>
Alexey
http://mlwiki.org/index.php?title=L%27Hopital%27s_Rule&diff=743&oldid=prev
Alexey at 06:26, 7 December 2015
2015-12-07T06:26:02Z
<p></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 06:26, 7 December 2015</td>
</tr><tr><td colspan="2" class="diff-lineno" id="L15" >Line 15:</td>
<td colspan="2" class="diff-lineno">Line 15:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>** know that $f(a) = g(a) = 0$, so have  </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>** know that $f(a) = g(a) = 0$, so have  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a)\, (x - a) + \ ...}{g'(a)\, (x - a) + \ ...}$</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a)\, (x - a) + \ ...}{g'(a)\, (x - a) + \ ...}$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* can factor $(x - a)$ out, so we have:  </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* can factor $(x - a)$ out, so <ins class="diffchange diffchange-inline">now </ins>we have:  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a) + \ ...}{g'(a) + \ ...}$</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a) + \ ...}{g'(a) + \ ...}$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* the leading order terms are $f'(a)$ and $g'(a)$, and the rest vanish under the limit</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* the leading order terms are $f'(a)$ and $g'(a)$, and the rest vanish under the limit</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="L39" >Line 39:</td>
<td colspan="2" class="diff-lineno">Line 39:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* so the limit is 0</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* so the limit is 0</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>To say that one function grows faster than other, we can use the [[<del class="diffchange diffchange-inline">Order </del>of Growth|Big-O notation]]</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>To say that one function grows faster than other, we can use the [[<ins class="diffchange diffchange-inline">Orders </ins>of Growth|Big-O notation]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Alexey
http://mlwiki.org/index.php?title=L%27Hopital%27s_Rule&diff=738&oldid=prev
Alexey: Created page with "== L'Hôpital's Rule == The rule has this form: $$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$ === $0/0$ case === * suppose $\lim\limits_{x \to..."
2015-12-06T20:48:39Z
<p>Created page with "== L'Hôpital's Rule == The rule has this form: $$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$ === $0/0$ case === * suppose $\lim\limits_{x \to..."</p>
<p><b>New page</b></p><div>== L'Hôpital's Rule ==<br />
The rule has this form:<br />
$$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$<br />
<br />
<br />
=== $0/0$ case ===<br />
* suppose $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \cfrac{0}{0}$<br />
* i.e. $f(a) = 0$ and $g(a) = 0$<br />
* if $f(x)$ and $g(x)$ are continuous <br />
* then the rule is $$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$<br />
<br />
<br />
Can show this using [[Taylor Expansion]] about $x = a$:<br />
* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f(a) + f'(a)\, (x - a) + \ ...}{g(a) + g'(a)\, (x - a) + \ ...}$<br />
** know that $f(a) = g(a) = 0$, so have <br />
* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a)\, (x - a) + \ ...}{g'(a)\, (x - a) + \ ...}$<br />
* can factor $(x - a)$ out, so we have: <br />
* $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \lim\limits_{x \to a} \cfrac{f'(a) + \ ...}{g'(a) + \ ...}$<br />
* the leading order terms are $f'(a)$ and $g'(a)$, and the rest vanish under the limit<br />
<br />
<br />
Examples:<br />
* $\lim\limits_{x \to 0} \cfrac{\sin x}{x} = \lim\limits_{x \to 0} \cfrac{\cos x}{1} = 1$<br />
* $\lim\limits_{x \to 0} \cfrac{1 - \cos x}{x} = \lim\limits_{x \to 0} \cfrac{\sin x}{1} = 0$<br />
<br />
<br />
=== $\infty / \infty$ case ===<br />
* suppose $\lim\limits_{x \to a} \cfrac{f(x)}{g(x)} = \cfrac{\infty}{\infty}$<br />
* i.e. $f(a) = \infty$ and $g(a) = \infty$<br />
* if $f(x)$ and $g(x)$ are continuous <br />
* then the rule is $$\lim_{x \to a} \cfrac{f(x)}{g(x)} = \lim_{x \to a} \cfrac{f'(x)}{g'(x)}$$<br />
<br />
<br />
Example: <br />
* $\lim\limits_{x \to \infty} \cfrac{\ln x}{\sqrt{x}}$<br />
* both go to $\infty$, but the rate at which they approach $\infty$ is different <br />
* by taking the derivative, we can see which one grows faster <br />
* is this case, $\sqrt{x}$ dominates $\ln x$: it grows much faster<br />
* so the limit is 0<br />
<br />
To say that one function grows faster than other, we can use the [[Order of Growth|Big-O notation]]<br />
<br />
<br />
=== Other Cases ===<br />
See http://calculus.seas.upenn.edu/?n=Main.LHopitalsRule<br />
<br />
<br />
== Sources ==<br />
* [[Calculus: Single Variable (coursera)]]<br />
<br />
[[Category:Calculus]]<br />
[[Category:Limits]]</div>
Alexey