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<h2 class="hd hd-2 unit-title">1. L'Hopital's Rule</h2>
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<p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Apply <b class="bf"><span style="color:#27408C">l'Hôpital's rule</span></b> to evaluate limits with <b class="bf"><span style="color:#27408C">indeterminate forms</span></b>: [mathjaxinline]\displaystyle \frac00, \frac{\infty }{\infty }[/mathjaxinline]. </p></li><li><p>
Rearrange limits of other indeterminate forms [mathjaxinline]\displaystyle 0\cdot \infty , \infty -\infty[/mathjaxinline] to be of the form [mathjaxinline]\displaystyle \frac00, \frac{\infty }{\infty }[/mathjaxinline]. </p></li><li><p>
Rearrange limits of the exponential indeterminate forms [mathjaxinline]0^0, 1^{\infty }, \infty ^0[/mathjaxinline] by applying [mathjaxinline]e^{\ln }[/mathjaxinline] to get a standard indeterminate form [mathjaxinline]\displaystyle \frac00, \frac{\infty }{\infty }[/mathjaxinline]. </p></li><li><p>
Use l'Hôpital's rule to <b class="bf"><span style="color:#27408C">compare the growth rates</span></b> of functions at infinity. </p></li></ul><p><b class="bfseries">Contents: 13 pages</b></p><p>
7 videos (31 minutes 1x speed) 20 questions </p>
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<h2 class="hd hd-2 unit-title">2. L'Hospital's Rule</h2>
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<h3 class="hd hd-2">L'Hospital's Rule</h3>
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If </p><table id="a0000000771" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\begin{array}{l} \displaystyle f(x) \rightarrow 0\\ \displaystyle g(x) \rightarrow 0 \end{array} \quad \textrm{ as } x \rightarrow a,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and the functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable near the point [mathjaxinline]x=a[/mathjaxinline], then limit </p><table id="a0000000772" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(2.1)</td></tr></table><p>
provided that the right hand limit exists or is [mathjaxinline]\pm \infty[/mathjaxinline]. </p><p>
If </p><table id="a0000000773" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\begin{array}{l} \displaystyle f(x) \rightarrow \pm \infty \\ \displaystyle g(x) \rightarrow \pm \infty \end{array} \quad \textrm{ as } x \rightarrow a,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and the functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable near the point [mathjaxinline]x=a[/mathjaxinline], then limit </p><table id="a0000000774" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(2.2)</td></tr></table><p>
provided that the right hand limit exists or is [mathjaxinline]\pm \infty[/mathjaxinline]. </p><p>
Note that </p><ul class="itemize"><li><p>
We can replace [mathjaxinline]a[/mathjaxinline] with [mathjaxinline]a^+[/mathjaxinline] or [mathjaxinline]a^-[/mathjaxinline] and the results (versions 1 and 2) still hold. </p></li><li><p>
We can replace [mathjaxinline]a[/mathjaxinline] with [mathjaxinline]\pm \infty[/mathjaxinline], and the results (versions 1 and 2) still hold. </p></li></ul>
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<h2 class="hd hd-2 unit-title">3. Practice</h2>
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<h3 class="hd hd-2">L'Hospital Practice</h3>
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Compute [mathjaxinline]\displaystyle \lim _{x\rightarrow \infty } \frac{\ln (x^2) + 4x + 30}{8x + \sqrt {x}}[/mathjaxinline]. </p>
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(Enter DNE if the limit does not exist and is not [mathjaxinline]\pm \infty[/mathjaxinline]. Enter infty if the limit is [mathjaxinline]\infty[/mathjaxinline], and -infty if the limit is [mathjaxinline]-\infty[/mathjaxinline].) </p>
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<h2 class="hd hd-2 unit-title">4. Justification of a special case of l'Hospital rule Version 1</h2>
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<p>
Here we give a different justification that the limit [mathjaxinline]\displaystyle \lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}[/mathjaxinline] in the special case of version 1 where [mathjaxinline]\ f(a) = g(a) = 0[/mathjaxinline], and [mathjaxinline]\ f'(a)[/mathjaxinline] and [mathjaxinline]\ g'(a)[/mathjaxinline] exist. </p><p>
Note that for [mathjaxinline]x[/mathjaxinline] near [mathjaxinline]a[/mathjaxinline], we can approximate [mathjaxinline]\ f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] by their linear approximations. This gives us: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{f(x)}{g(x)}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \approx \frac{f(a)+f'(a)\Delta x}{g(a)+g'(a)\Delta x}[/mathjaxinline] </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:right; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{f'(a)\Delta x}{g'(a)\Delta x}[/mathjaxinline] </td><td style="text-align:left; border:none">
since [mathjaxinline]\ f(a) = g(a) = 0[/mathjaxinline] </td></tr><tr><td style="text-align:right; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{f'(a)}{g'(a)}[/mathjaxinline] </td><td style="text-align:left; border:none">
provided [mathjaxinline]\ f'(a)[/mathjaxinline] and [mathjaxinline]g'(a)[/mathjaxinline] exist. </td></tr></table><p>
What if in both [mathjaxinline]\displaystyle f(a) = g(a) = 0[/mathjaxinline] and [mathjaxinline]\displaystyle f'(a) = g'(a) = 0[/mathjaxinline]? Use the quadratic approximation and see what you get. </p>
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<h2 class="hd hd-2 unit-title">5. Applying l'Hospital repeatedly</h2>
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Practice applying rule repeatedly
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Calculate [mathjaxinline]\displaystyle \lim _{x\rightarrow 1} \frac{e^ x - ex}{x^3 - x^2 - x +1}[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">6. Exponential Function Asymptotics</h2>
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Compute [mathjaxinline]\displaystyle {\lim _{x\rightarrow \infty } \frac{x^{1000000}}{e^ x}}[/mathjaxinline]. </p>
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(Enter DNE if the limit does not exist and is not [mathjaxinline]\pm \infty[/mathjaxinline]. Enter infty if the limit is [mathjaxinline]\infty[/mathjaxinline], and -infty if the limit is [mathjaxinline]-\infty[/mathjaxinline].) </p>
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Exponential versus polynomial growth rate
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<p>
Compare [mathjaxinline]x^ n[/mathjaxinline] and [mathjaxinline]e^ x[/mathjaxinline] where [mathjaxinline]n[/mathjaxinline] is a positive integer. Which function grows faster as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]+\infty[/mathjaxinline]? </p>
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<text> [mathjaxinline]x^ n[/mathjaxinline]</text>
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<text> We cannot tell.</text>
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Logarithm growth rate
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Compare [mathjaxinline]\ln x[/mathjaxinline] to [mathjaxinline]\sqrt [n]{x}[/mathjaxinline], where [mathjaxinline]n[/mathjaxinline] is a positive integer. Which function grows faster as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]+\infty[/mathjaxinline]? </p>
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<text> [mathjaxinline]\sqrt [n]{x}[/mathjaxinline]</text>
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<text> [mathjaxinline]\ln x[/mathjaxinline]</text>
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<text> They will grow at the same rate</text>
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<h2 class="hd hd-2 unit-title">7. Other Indeterminate Forms</h2>
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<p>
Recall that [mathjaxinline]\displaystyle \frac00[/mathjaxinline] and [mathjaxinline]\displaystyle \frac{\infty }{\infty }[/mathjaxinline] are <b class="bf"><span style="color:#27408C">indeterminate forms.</span></b> </p><p>
There are other indeterminate forms. Let's figure out what they are. </p>
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Indeterminate Product 1?
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<p>
Suppose </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000801" style="table-layout:auto" width="100%">
<tr id="a0000000802">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} f(x) = 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.22)</td>
</tr>
<tr id="a0000000803">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} g(x) = +\infty .[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.23)</td>
</tr>
</table>
<p>
What is [mathjaxinline]\displaystyle \ \lim _{x\rightarrow a} f(x)g(x)[/mathjaxinline]? </p>
<p>
(If you're not sure, make a tentative guess and then test out what happens when you multiply really small numbers by really big numbers. For instance, try [mathjaxinline]10^{-6} \times 10^{10}[/mathjaxinline] or [mathjaxinline]10^{-10} \times 10^3[/mathjaxinline].) </p>
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<text> It is close to zero.</text>
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<text> It is close to 1.</text>
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<text> It is really large and positive.</text>
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<text> It is really large and negative.</text>
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<text> We can't tell.</text>
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Indeterminate Product 2?
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Suppose </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000804" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \lim _{x\rightarrow a} f(x) = \lim _{x\rightarrow a} g(x) = +\infty .[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
What is [mathjaxinline]\displaystyle \lim _{x\rightarrow a} f(x)g(x)[/mathjaxinline]? </p>
<p>
(Test out what happens when you multiply really big numbers by really big numbers.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lhop-tab7-problem2_2_1">
<fieldset aria-describedby="status_lhop-tab7-problem2_2_1">
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<input type="radio" name="input_lhop-tab7-problem2_2_1" id="input_lhop-tab7-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lhop-tab7-problem2_2_1-choice_1-label" for="input_lhop-tab7-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem2_2_1">
<text> It is close to zero.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lhop-tab7-problem2_2_1" id="input_lhop-tab7-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lhop-tab7-problem2_2_1-choice_2-label" for="input_lhop-tab7-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem2_2_1">
<text> It is close to 1.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lhop-tab7-problem2_2_1" id="input_lhop-tab7-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lhop-tab7-problem2_2_1-choice_3-label" for="input_lhop-tab7-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem2_2_1">
<text> It is really large and positive.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lhop-tab7-problem2_2_1" id="input_lhop-tab7-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lhop-tab7-problem2_2_1-choice_4-label" for="input_lhop-tab7-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem2_2_1">
<text> It is really large and negative.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lhop-tab7-problem2_2_1" id="input_lhop-tab7-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lhop-tab7-problem2_2_1-choice_5-label" for="input_lhop-tab7-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem2_2_1">
<text> We can't tell.</text>
</label>
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Indeterminate Difference?
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Suppose [mathjaxinline]\displaystyle \lim _{x\rightarrow a} f(x) = \lim _{x\rightarrow a} g(x) = +\infty .[/mathjaxinline] </p>
<p>
What is [mathjaxinline]\displaystyle \lim _{x\rightarrow a}\left( f(x)-g(x)\right)[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lhop-tab7-problem3_2_1">
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<input type="radio" name="input_lhop-tab7-problem3_2_1" id="input_lhop-tab7-problem3_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lhop-tab7-problem3_2_1-choice_1-label" for="input_lhop-tab7-problem3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem3_2_1">
<text> It is close to zero.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lhop-tab7-problem3_2_1" id="input_lhop-tab7-problem3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lhop-tab7-problem3_2_1-choice_2-label" for="input_lhop-tab7-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem3_2_1">
<text> It is close to 1.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lhop-tab7-problem3_2_1" id="input_lhop-tab7-problem3_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lhop-tab7-problem3_2_1-choice_3-label" for="input_lhop-tab7-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem3_2_1">
<text> It is really large and positive.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lhop-tab7-problem3_2_1" id="input_lhop-tab7-problem3_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lhop-tab7-problem3_2_1-choice_4-label" for="input_lhop-tab7-problem3_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem3_2_1">
<text> It is really large and negative.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lhop-tab7-problem3_2_1" id="input_lhop-tab7-problem3_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lhop-tab7-problem3_2_1-choice_5-label" for="input_lhop-tab7-problem3_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lhop-tab7-problem3_2_1">
<text> We can't tell.</text>
</label>
</div>
<span id="answer_lhop-tab7-problem3_2_1"/>
</fieldset>
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<h2 class="hd hd-2 unit-title">8. Other Indeterminate Forms</h2>
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<h3 class="hd hd-2">Dealing with other indeterminate forms</h3>
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<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_lhop-tab8-video1">Downloads and transcripts</h3>
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<h4 class="hd hd-5">Transcripts</h4>
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More Practice 1
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<p>
Calculate [mathjaxinline]\displaystyle \lim _{x\rightarrow 0^+} xe^{1/x}[/mathjaxinline]. </p>
<p>
(Enter DNE if the limit does not exist and is not [mathjaxinline]\pm \infty[/mathjaxinline]. Enter infty if the limit is [mathjaxinline]\infty[/mathjaxinline], and -infty if the limit is [mathjaxinline]-\infty[/mathjaxinline].) </p>
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More Practice 2
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Calculate [mathjaxinline]\displaystyle \lim _{x\rightarrow \pi /2} \sec x - \tan x[/mathjaxinline]. </p>
<p>
(Enter DNE if the limit does not exist and is not [mathjaxinline]\pm \infty[/mathjaxinline]. Enter infty if the limit is [mathjaxinline]\infty[/mathjaxinline], and -infty if the limit is [mathjaxinline]-\infty[/mathjaxinline].) </p>
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<h2 class="hd hd-2 unit-title">9. Exponential Indeterminate Forms</h2>
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Indeterminate Exponential 1?
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<p>
The trickiest <b class="bf"><span style="color:#99182C">indeterminate forms</span></b> are ones involving exponents. </p>
<p>
Suppose </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000813" style="table-layout:auto" width="100%">
<tr id="a0000000814">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} f(x)[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle = 1[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.29)</td>
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<tr id="a0000000815">
<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} g(x)[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle = +\infty .[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.30)</td>
</tr>
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<p>
What is [mathjaxinline]\displaystyle \lim _{x \rightarrow a} f(x)^{g(x)}[/mathjaxinline]? </p>
<p>
(If you're not sure, try one of the following. First, think about what each part is trying to do to the overall result - are they working together or against each other? Alternatively, plug in some numbers: [mathjaxinline]1.0001^{10^6}[/mathjaxinline], [mathjaxinline].9995^{10^6}[/mathjaxinline].) </p>
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Indeterminate Exponential 2?
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Assume that [mathjaxinline]\ f(x) \geq 0[/mathjaxinline], and that </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000817" style="table-layout:auto" width="100%">
<tr id="a0000000818">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} f(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle = 0[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.31)</td>
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<tr id="a0000000819">
<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} g(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle = +\infty .[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.32)</td>
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What is [mathjaxinline]\displaystyle \lim _{x\rightarrow a} f(x)^{g(x)}[/mathjaxinline]? </p>
<p>
(If you're not sure, try one of the following. First, think about what each part is trying to do to the overall result - are they working together or against each other? Alternatively, plug in some numbers: [mathjaxinline]\left(10^{-10}\right)^{10^10}.[/mathjaxinline]) </p>
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Indeterminate Exponential 3?
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The trickiest indeterminate forms are ones involving exponents. </p>
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Suppose </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000821" style="table-layout:auto" width="100%">
<tr id="a0000000822">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} f(x)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = +\infty[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.33)</td>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} g(x)[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle = 0.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.34)</td>
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<p>
What is [mathjaxinline]\displaystyle \lim _{x\rightarrow a}f(x)^{g(x)}[/mathjaxinline]? </p>
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Indeterminate Exponential 4?
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Assume that [mathjaxinline]\ f(x) \geq 0[/mathjaxinline] and that </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000824" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} f(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle = 0[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.35)</td>
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<tr id="a0000000826">
<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle \lim _{x\rightarrow a} g(x)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 0.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.36)</td>
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<p>
What is [mathjaxinline]\displaystyle \lim _{x\rightarrow a}f(x)^{g(x)}[/mathjaxinline]? </p>
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<text> It is really large and positive.</text>
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<h2 class="hd hd-2 unit-title">10. Dealing with Exponential Indeterminate Forms</h2>
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The forms [mathjaxinline]0^0[/mathjaxinline], [mathjaxinline]\infty ^0[/mathjaxinline], and [mathjaxinline]1^{\infty }[/mathjaxinline] are all indeterminate forms. To apply l'Hôpital's rule, we must rearrange these expressions so that they look like [mathjaxinline]\displaystyle \frac00[/mathjaxinline] or [mathjaxinline]\displaystyle \frac{\infty }{\infty }[/mathjaxinline]. The trick here is to change to base [mathjaxinline]e[/mathjaxinline]. </p>
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<h3 class="hd hd-2">Dealing with exponential indeterminate forms</h3>
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Calculate [mathjaxinline]\displaystyle \lim _{x\rightarrow \pi /2^-} \left( \frac{2x}{\pi } \right)^{\tan x}[/mathjaxinline]. </p>
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(Type e for Euler's number and type pi for [mathjaxinline]\pi[/mathjaxinline]. Enter <b class="bf">DNE</b> if the limit does not exist and is not [mathjaxinline]\pm \infty[/mathjaxinline]. Enter <b class="bf">infty </b> if the limit is [mathjaxinline]\infty[/mathjaxinline], and <b class="bf">-infty </b> if the limit is [mathjaxinline]-\infty[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \lim _{n \rightarrow \infty } \left( 1+ \frac1{n} \right)^{n}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \lim _{n \rightarrow \infty } \left( 1- \frac1{n} \right)^{n}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">11. Look before you L'Hop</h2>
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<p>
The title of this video is a pun on the expression: </p><p><i class="itshape">Look before you leap</i>, </p><p>
which means to think about what you are doing. Here the same reasoning applies. Stop and think about what you are doing before you blindly apply L'Hôpital everywhere! </p>
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<h2 class="hd hd-2 unit-title">12. Summary</h2>
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<p><b class="bfseries">l'Hôpital's Rule</b></p><p><b class="bfseries">L'Hôpital's Rule Version 1: Indeterminate form [mathjaxinline]\displaystyle \frac00[/mathjaxinline] </b></p><p>
If </p><table id="a0000000843" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\begin{array}{l} \displaystyle f(x) \rightarrow 0\\ \displaystyle g(x) \rightarrow 0 \end{array} \quad \textrm{ as } x \rightarrow a,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and the functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable near the point [mathjaxinline]x=a[/mathjaxinline], then limit </p><table id="a0000000844" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(2.48)</td></tr></table><p>
provided that the right hand limit exists or is [mathjaxinline]\pm \infty[/mathjaxinline]. </p><p><b class="bfseries">L'Hôpital's Rule Version 2: Indeterminate form [mathjaxinline]\displaystyle \frac{\infty }{\infty }[/mathjaxinline] </b></p><p>
If </p><table id="a0000000845" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\begin{array}{l} \displaystyle f(x) \rightarrow \pm \infty \\ \displaystyle g(x) \rightarrow \pm \infty \end{array} \quad \textrm{ as } x \rightarrow a,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and the functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline] are differentiable near the point [mathjaxinline]x=a[/mathjaxinline], then limit </p><table id="a0000000846" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none;text-align:right">(2.49)</td></tr></table><p>
provided that the right hand limit exists or is [mathjaxinline]\pm \infty[/mathjaxinline]. </p><p>
Note that </p><ul class="itemize"><li><p>
We can replace [mathjaxinline]a[/mathjaxinline] with [mathjaxinline]a^+[/mathjaxinline] or [mathjaxinline]a^-[/mathjaxinline] and the results (versions 1 and 2) still hold. </p></li><li><p>
We can replace [mathjaxinline]a[/mathjaxinline] with [mathjaxinline]\pm \infty[/mathjaxinline], and the results (versions 1 and 2) still hold. </p></li></ul><p><br/></p><p><b class="bfseries">Other indeterminate forms</b></p><p>
Other indeterminate forms [mathjaxinline]0\cdot \infty[/mathjaxinline], [mathjaxinline]\infty - \infty[/mathjaxinline], [mathjaxinline]0^0[/mathjaxinline], [mathjaxinline]1^{\infty }[/mathjaxinline], and [mathjaxinline]\infty ^{0}[/mathjaxinline] should be rearranged to be of the form [mathjaxinline]0/0[/mathjaxinline] or [mathjaxinline]\infty /\infty[/mathjaxinline] in order to apply l'Hôpital's rule. </p>
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<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.3x+1T2020+type@html+block@lhop-tab12-text2" data-course-id="course-v1:MITx+18.01.3x+1T2020" data-has-score="False" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-request-token="ebc16f92036611efa53c0afff417eba9">
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<p>
For a printable version, go <a href="https://courses.edx.org/asset-v1:MITx+18.01.3x+1T2020+type@asset+block@lhop_summary.pdf" target="_blank">here</a>. </p>
</div>
</div>
</div>
</div>
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