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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">The Big Picture</h3>
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<h2 class="hd hd-2 unit-title">2. Limits and asymptotics</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>
Use the general strategy for <b class="bf"><span style="color:#27408C">sketching curves by hand</span></b>. </p></li><li><p>
Use limits to evaluate function values at <b class="bf"><span style="color:#27408C">discontinuities </span></b> and <b class="bf"><span style="color:#27408C">end points</span></b> (including [mathjaxinline]\pm \infty[/mathjaxinline]). </p></li><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: 18 pages</b></p><p>
13 videos (56 minutes 1x speed) 27 questions </p>
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<h2 class="hd hd-2 unit-title">3. Graphing</h2>
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Review: Evaluate the right and left limits
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We introduced the notation [mathjaxinline]\displaystyle \ f(-2^{{}+{}}) = \lim _{x \rightarrow -2^{{}+{}}} f(x)[/mathjaxinline]. Note that [mathjaxinline]-2^{{}+{}} = (-2)^{{}+{}}[/mathjaxinline]. Similarly we let [mathjaxinline]\displaystyle \ f(-2^{{}-{}})[/mathjaxinline] denote [mathjaxinline]\ \lim _{x \rightarrow -2^{{}-{}}} f(x)[/mathjaxinline]. Note that [mathjaxinline]-2^{{}-{}} = (-2)^{{}-{}}[/mathjaxinline]. </p>
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Evaluate the right and left hand limits of the function [mathjaxinline]\displaystyle \frac{x+1}{x+2}[/mathjaxinline] at the discontinuity [mathjaxinline]x=-2[/mathjaxinline]. </p>
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(Type infty for [mathjaxinline]+\infty[/mathjaxinline] and -infty for [mathjaxinline]-\infty[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]\ f(-2^{{}+{}}) =[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\ f(-2^{{}-{}}) =[/mathjaxinline]</p>
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<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_app_2-tab3-video2">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab3-video2/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
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<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab3-video2/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
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</div>
</div>
</div>
</div>
</div>
</div>
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<h2 class="hd hd-2 unit-title">4. General Strategy and worked example</h2>
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<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<ol class="enumerate"><li value="1"><p>
Plot </p><ul class="itemize"><li><p>
discontinuities (especially infinite ones) </p></li><li><p>
end points (or [mathjaxinline]x \rightarrow \pm \infty[/mathjaxinline]) </p></li><li><p>
easy points ([mathjaxinline]x = 0[/mathjaxinline], or [mathjaxinline]y=0[/mathjaxinline]) (This is optional.) </p></li></ul></li><li value="2"><p>
Plot critical points and values. (Solve [mathjaxinline]\ f'(x) = 0[/mathjaxinline] or undefined.) </p></li><li value="3"><p>
Decide whether [mathjaxinline]\ f' < 0[/mathjaxinline] or [mathjaxinline]\ f' >0[/mathjaxinline] on each interval between endpoints, critical points, and discontinuities. (Valuable double check) </p></li><li value="4"><p>
Identify where [mathjaxinline]\ f^{\prime \prime } < 0[/mathjaxinline] and [mathjaxinline]\ f^{\prime \prime } > 0[/mathjaxinline] (concave down and concave up). </p><p>
Identify inflection points. (Makes graph look nice. Can be used to double check.) </p></li><li value="5"><p>
Combine into graph. </p></li></ol>
</div>
</div>
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<h2 class="hd hd-2 unit-title">5. Example</h2>
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Worked example: Step 1
</h3>
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<p>
Consider the function [mathjaxinline]\displaystyle \frac{x}{(\ln x)^2}[/mathjaxinline]. First step. Identify the discontinuities and the right and left hand limit at the discontinuity. </p>
<p>
(Type <b class="bf">infty</b> for [mathjaxinline]\infty[/mathjaxinline] and <b class="bf">-infty</b> for [mathjaxinline]-\infty[/mathjaxinline].) </p>
<p>
<p style="display:inline">Discontinuity: [mathjaxinline]x=[/mathjaxinline]</p>
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<p>
<p style="display:inline">Righthand limit at discontinuity =</p>
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\(\)
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<br/>
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<p>
<p style="display:inline">Lefthand limit at discontinuity =</p>
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Worked example: identify end points
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Consider the function [mathjaxinline]\displaystyle \ f(x) = \frac{x}{(\ln x)^2}[/mathjaxinline]. Next identify the right and left endpoints. </p>
<p>
(Type <b class="bf">infty</b> for [mathjaxinline]\infty[/mathjaxinline]; <b class="bf">-infty</b> for [mathjaxinline]-\infty[/mathjaxinline].) </p>
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<p style="display:inline">Left endpoint: [mathjaxinline]x=[/mathjaxinline]</p>
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<p style="display:inline">Right endpoint: [mathjaxinline]x=[/mathjaxinline]</p>
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<h3 class="hd hd-2">Evaluating the end points</h3>
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We call [mathjaxinline]\displaystyle \frac00[/mathjaxinline] and [mathjaxinline]\displaystyle \frac{\infty }{\infty }[/mathjaxinline] <b class="bf"><span style="color:#27408C">indeterminate forms,</span></b> because when we run into them in a limit, they require further analysis to determine whether the numerator or denominator wins in the race to 0 or [mathjaxinline]\infty[/mathjaxinline] respectively, or whether they balance out and reach some other finite limit. </p>
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<h2 class="hd hd-2 unit-title">6. Critical points of graph</h2>
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Identify critical points
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We are graphing the function [mathjaxinline]\ \displaystyle f(x) = \frac{x}{(\ln x)^2}[/mathjaxinline]. The derivative is [mathjaxinline]\ \displaystyle f'(x) = \frac{\ln x - 2}{(\ln x)^3}[/mathjaxinline]. </p>
<p>
Identify the points where [mathjaxinline]\ f'(x)=0[/mathjaxinline]. </p>
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(If there is more than one, separate multiple entries with a comma: e.g. pi, 2.3. If there is no critical point on the designated interval, type none.) </p>
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Intervals where increasing and decreasing
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Identify the intervals where [mathjaxinline]\ f'(x)[/mathjaxinline] is positive and negative. </p>
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(Enter intervals between parentheses; e.g. (a, b). If there is more than one interval, separate with commas; e.g. (a, b), (c, d). Type <b class="bf">infty</b> for [mathjaxinline]\infty[/mathjaxinline] and <b class="bf">-infty</b> for [mathjaxinline]-\infty[/mathjaxinline]. You may type e for Euler's number, and pi for the mathematical constant [mathjaxinline]\pi[/mathjaxinline].) </p>
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<p style="display:inline">Interval(s) where [mathjaxinline]\ f' &gt;0[/mathjaxinline]: </p>
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<h2 class="hd hd-2 unit-title">7. Sketch the function</h2>
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Now it is your turn to try sketching some functions. First try the recitation problem posed in the video, and next you work through a problem on your own. </p>
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Sketch the function
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<p>
Sketch the rational function [mathjaxinline]\ f(x)[/mathjaxinline] defined below. </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000672" 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 f(x) = \frac{2x^2+1}{x^2-1}[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(6.6)</td>
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<tr id="a0000000674">
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[mathjaxinline]\displaystyle \displaystyle f'(x) = \frac{-6x}{(x^2-1)^2}[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(6.7)</td>
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[mathjaxinline]\displaystyle \displaystyle f^{\prime \prime }(x) = \frac{6(3x^2+1)}{(x^2-1)^3}[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(6.8)</td>
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<p>
(The locations of asymptotes should be exact. The function sketch can be qualitative. Use the point tool to label any extrema and any inflection points.) </p>
<font size="2">Note that when drawing the function, you may choose between using the freeform or spline drawing tool using the dropdown menu in the menu of the sketch response tool. The freeform tool draws like a regular pencil using your mouse. The spline tool allows you to pick a series of discrete points, and it then connects them (once you have two or more points) with a curve. <br/> <br/> Note an Ungraded Point tool is provided to help your sketch.</font>
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<h2 class="hd hd-2 unit-title">8. L'Hospital's Rule</h2>
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<p><b class="bfseries">L'Hôpital's Rule Version 1: Indeterminate form [mathjaxinline]\displaystyle \frac00[/mathjaxinline] </b></p><p>
If </p><table id="a0000000682" 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 \text { 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="a0000000683" 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">(6.13)</td></tr></table><p>
provided that the limit on the right hand side of the equation 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="a0000000684" 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 \text { 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="a0000000685" 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">(6.14)</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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Compute [mathjaxinline]\displaystyle {\lim _{x\rightarrow \infty } \frac{\ln (e^ x+5x)}{4x +3}}[/mathjaxinline]. </p>
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(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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<h2 class="hd hd-2 unit-title">10. Justification of a special case of l'Hospital rule Version 1</h2>
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Here we give a 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 the right hand limit exists. </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">11. Applying l'Hospital repeatedly</h2>
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Applying rule repeatedly practice
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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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(Type e for Euler's number. 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]. You may type e into the answer box for Euler's number.) </p>
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<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
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Exponential Ratio
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Compute [mathjaxinline]\displaystyle \lim _{x\rightarrow \infty } \frac{300x^3 + 17x}{e^ x}[/mathjaxinline]. </p>
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(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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Exponential Ratio 2
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Compute [mathjaxinline]\displaystyle {\lim _{x\rightarrow \infty } \frac{x^{1000000}}{e^ x}}[/mathjaxinline]. </p>
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(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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Exponential versus polynomial growth rate
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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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Logarithm growth rate
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<p>
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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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_2-tab11-problem5_2_1">
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<text> [mathjaxinline]\sqrt [n]{x}[/mathjaxinline]</text>
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<input type="radio" name="input_app_2-tab11-problem5_2_1" id="input_app_2-tab11-problem5_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_2-tab11-problem5_2_1-choice_2-label" for="input_app_2-tab11-problem5_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_2-tab11-problem5_2_1">
<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">12. Other Indeterminate Forms</h2>
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<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="57d0c961ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@app_2-tab12-text1" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
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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?
</h3>
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<p>
Suppose </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000712" style="table-layout:auto" width="100%">
<tr id="a0000000713">
<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">(6.34)</td>
</tr>
<tr id="a0000000714">
<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">(6.35)</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 (positive) numbers by really big (positive) numbers. For instance, try [mathjaxinline]10^{-6} \times 10^{10}[/mathjaxinline] or [mathjaxinline]10^{-10} \times 10^3[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_2-tab12-problem1_2_1">
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<text> It is close to zero.</text>
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<input type="radio" name="input_app_2-tab12-problem1_2_1" id="input_app_2-tab12-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_2-tab12-problem1_2_1-choice_2-label" for="input_app_2-tab12-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem1_2_1">
<text> It is close to 1.</text>
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<input type="radio" name="input_app_2-tab12-problem1_2_1" id="input_app_2-tab12-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="app_2-tab12-problem1_2_1-choice_3-label" for="input_app_2-tab12-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem1_2_1">
<text> It is really large and positive.</text>
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<input type="radio" name="input_app_2-tab12-problem1_2_1" id="input_app_2-tab12-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="app_2-tab12-problem1_2_1-choice_4-label" for="input_app_2-tab12-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem1_2_1">
<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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<p>
Suppose </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000715" 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 postive numbers by really big positive numbers.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_2-tab12-problem2_2_1">
<fieldset aria-describedby="status_app_2-tab12-problem2_2_1">
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<input type="radio" name="input_app_2-tab12-problem2_2_1" id="input_app_2-tab12-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="app_2-tab12-problem2_2_1-choice_1-label" for="input_app_2-tab12-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem2_2_1">
<text> It is close to zero.</text>
</label>
</div>
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<input type="radio" name="input_app_2-tab12-problem2_2_1" id="input_app_2-tab12-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_2-tab12-problem2_2_1-choice_2-label" for="input_app_2-tab12-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem2_2_1">
<text> It is close to 1.</text>
</label>
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<input type="radio" name="input_app_2-tab12-problem2_2_1" id="input_app_2-tab12-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="app_2-tab12-problem2_2_1-choice_3-label" for="input_app_2-tab12-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem2_2_1">
<text> It is really large and positive.</text>
</label>
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<div class="field">
<input type="radio" name="input_app_2-tab12-problem2_2_1" id="input_app_2-tab12-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="app_2-tab12-problem2_2_1-choice_4-label" for="input_app_2-tab12-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem2_2_1">
<text> It is really large and negative.</text>
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<input type="radio" name="input_app_2-tab12-problem2_2_1" id="input_app_2-tab12-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="app_2-tab12-problem2_2_1-choice_5-label" for="input_app_2-tab12-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem2_2_1">
<text> We can't tell.</text>
</label>
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<span class="status unanswered" id="status_app_2-tab12-problem2_2_1" data-tooltip="Not yet answered.">
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<h3 class="hd hd-3 problem-header" id="app_2-tab12-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab12-problem3-problem-progress" tabindex="-1">
Indeterminate Difference?
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab12-problem3-problem-progress"></div>
<div class="problem">
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<p>
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_app_2-tab12-problem3_2_1">
<fieldset aria-describedby="status_app_2-tab12-problem3_2_1">
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<input type="radio" name="input_app_2-tab12-problem3_2_1" id="input_app_2-tab12-problem3_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="app_2-tab12-problem3_2_1-choice_1-label" for="input_app_2-tab12-problem3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem3_2_1">
<text> It is close to zero.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_2-tab12-problem3_2_1" id="input_app_2-tab12-problem3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_2-tab12-problem3_2_1-choice_2-label" for="input_app_2-tab12-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem3_2_1">
<text> It is close to 1.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_2-tab12-problem3_2_1" id="input_app_2-tab12-problem3_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="app_2-tab12-problem3_2_1-choice_3-label" for="input_app_2-tab12-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem3_2_1">
<text> It is really large and positive.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_2-tab12-problem3_2_1" id="input_app_2-tab12-problem3_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="app_2-tab12-problem3_2_1-choice_4-label" for="input_app_2-tab12-problem3_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem3_2_1">
<text> It is really large and negative.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_app_2-tab12-problem3_2_1" id="input_app_2-tab12-problem3_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="app_2-tab12-problem3_2_1-choice_5-label" for="input_app_2-tab12-problem3_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_app_2-tab12-problem3_2_1">
<text> We can't tell.</text>
</label>
</div>
<span id="answer_app_2-tab12-problem3_2_1"/>
</fieldset>
<div class="indicator-container">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</div>
</div></div>
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<h2 class="hd hd-2 unit-title">13. Other Indeterminate Forms</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab13-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="57d0c961ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab13-video1" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="video" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
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<h3 class="hd hd-2">Dealing with other indeterminate forms</h3>
<div
id="video_app_2-tab13-video1"
class="video closed"
data-metadata='{"saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab13-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "ytTestTimeout": 1500, "autohideHtml5": false, "completionPercentage": 0.95, "poster": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "sources": [], "transcriptLanguages": {"en": "English"}, "saveStateEnabled": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "autoAdvance": false, "duration": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab13-video1/handler/publish_completion", "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab13-video1/handler/transcript/translation/__lang__", "prioritizeHls": false, "autoplay": false, "savedVideoPosition": 0.0, "streams": "1.00:OVx_FA6uUjY", "speed": null, "generalSpeed": 1.0, "ytMetadataEndpoint": "", "showCaptions": "true", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "end": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab13-video1/handler/transcript/available_translations"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
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<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
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<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
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<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_app_2-tab13-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_app_2-tab13-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab13-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
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<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_2-tab13-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
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data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab13-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab13-problem1/handler/xmodule_handler"
data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="app_2-tab13-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab13-problem1-problem-progress" tabindex="-1">
More Practice 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab13-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Calculate [mathjaxinline]\displaystyle \lim _{x\rightarrow 0^+} xe^{1/x}[/mathjaxinline]. </p>
<p>
(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>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_app_2-tab13-problem1_2_1" class="inputtype formulaequationinput">
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</span>
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\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_app_2-tab13-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
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data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab13-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab13-problem2/handler/xmodule_handler"
data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="app_2-tab13-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab13-problem2-problem-progress" tabindex="-1">
More Practice 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_2-tab13-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Calculate [mathjaxinline]\displaystyle \lim _{x\rightarrow \pi /2} \sec x - \tan x[/mathjaxinline]. </p>
<p>
(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>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_app_2-tab13-problem2_2_1" class="inputtype formulaequationinput">
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<span class="trailing_text" id="trailing_text_app_2-tab13-problem2_2_1"/>
<span class="status unanswered" id="status_app_2-tab13-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_app_2-tab13-problem2_2_1" class="answer"/>
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\(\)
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<h2 class="hd hd-2 unit-title">14. 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="a0000000723" style="table-layout:auto" width="100%">
<tr id="a0000000724">
<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 = 1[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(6.40)</td>
</tr>
<tr id="a0000000725">
<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">(6.41)</td>
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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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<text> It is close to zero.</text>
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<text> It is really large and positive.</text>
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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="a0000000727" style="table-layout:auto" width="100%">
<tr id="a0000000728">
<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 style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(6.42)</td>
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<tr id="a0000000729">
<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} g(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle = +\infty .[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(6.43)</td>
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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]\left(10^{-10}\right)^{10^{10}}.[/mathjaxinline]) </p>
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<text> It is close to zero.</text>
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Indeterminate Exponential 3?
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The trickiest indeterminate forms are ones involving exponents. </p>
<p>
Suppose </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000731" style="table-layout:auto" width="100%">
<tr id="a0000000732">
<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]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(6.44)</td>
</tr>
<tr id="a0000000733">
<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} g(x)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = 0.[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(6.45)</td>
</tr>
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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>
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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">(6.46)</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>
<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">(6.47)</td>
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<p>
What is [mathjaxinline]\displaystyle \lim _{x\rightarrow a}f(x)^{g(x)}[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">15. 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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More Practice
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Calculate [mathjaxinline]\displaystyle \lim _{x\rightarrow \pi /2^-} \left( \frac{2x}{\pi } \right)^{\tan x}[/mathjaxinline]. </p>
<p>
(Type e for Euler's number and type pi for [mathjaxinline]\pi[/mathjaxinline]. Enter DNE 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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<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput">Fractions</td>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Finding e
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Calculate the following limits. </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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<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">17. Sketching problem</h2>
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Sketch [mathjaxinline]\displaystyle \frac{e^{x}}{x}[/mathjaxinline] on the interval [mathjaxinline]-\infty &lt; x &lt; \infty[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">18. Summary</h2>
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<p><b class="bfseries">Curve Sketching</b></p><p><b class="bfseries">General Strategy</b></p><ol class="enumerate"><li value="1"><p>
Plot </p><ul class="itemize"><li><p>
discontinuities (especially infinite ones) </p></li><li><p>
end points (or [mathjaxinline]x \rightarrow \pm \infty[/mathjaxinline]) </p></li><li><p>
easy points ([mathjaxinline]x = 0[/mathjaxinline], or [mathjaxinline]y=0[/mathjaxinline]) (This is optional.) </p></li></ul></li><li value="2"><p>
Plot critical points and values. (Solve [mathjaxinline]\ f'(x) = 0[/mathjaxinline] or undefined.) </p></li><li value="3"><p>
Decide whether [mathjaxinline]\ f' < 0[/mathjaxinline] or [mathjaxinline]\ f' >0[/mathjaxinline] on each interval between endpoints, critical points, and discontinuities. (Must be consistent with steps 1 and 2.) </p></li><li value="4"><p>
Identify where [mathjaxinline]\ f^{\prime \prime } < 0[/mathjaxinline] and [mathjaxinline]\ f^{\prime \prime } > 0[/mathjaxinline] (concave down and concave up). Identify inflection points. </p></li><li value="5"><p>
Combine into graph. </p></li></ol><p><br/></p><p><b class="bfseries">l'Hospital'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="a0000000760" 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 \text { 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="a0000000761" 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">(6.63)</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="a0000000762" 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 \text { 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="a0000000763" 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">(6.64)</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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