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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Introduction to Limits</h3>
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<h2 class="hd hd-2 unit-title">2. Introduction to limits</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 a calculator to determine right and left hand limits. </p></li><li><p>
Identify right and left hand limits based on graphs. </p></li><li><p>
Determine if a limit exists based on values of right and left hand limits. </p></li><li><p>
Understand that the limit does not depend on the value of a function at the point of interest. </p></li></ul><p><b class="bfseries">Contents: 14 pages</b></p><p>
6 videos (24 minutes 1x speed) 17 questions </p>
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<h2 class="hd hd-2 unit-title">5. Definitions of right-hand and left-hand limits</h2>
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Suppose [mathjaxinline]\ f(x)[/mathjaxinline] gets really close to [mathjaxinline]R[/mathjaxinline] for values of [mathjaxinline]x[/mathjaxinline] that get really close to (but are not equal to) [mathjaxinline]a[/mathjaxinline] from the right. Then we say [mathjaxinline]R[/mathjaxinline] is the <span style="color:#27408C"><b class="bf">right-hand limit</b></span> of the function [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from the right. </p><p>
We write </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:center; border:none">
[mathjaxinline]f(x) \rightarrow R[/mathjaxinline] as [mathjaxinline]x \rightarrow a^+[/mathjaxinline] </td></tr><tr><td style="text-align:center; border:none">
or </td></tr><tr><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow \mathbf{a^+}} f(x) = R}[/mathjaxinline]. </td></tr></table><p>
If [mathjaxinline]\ f(x)[/mathjaxinline] gets really close to [mathjaxinline]L[/mathjaxinline] for values of [mathjaxinline]x[/mathjaxinline] that get really close to (but are not equal to) [mathjaxinline]a[/mathjaxinline] from the left, we say that [mathjaxinline]L[/mathjaxinline] is the <span style="color:#27408C"><b class="bf">left-hand limit</b></span> of the function [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from the left. </p><p>
We write </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:center; border:none">
[mathjaxinline]f(x) \rightarrow L[/mathjaxinline] as [mathjaxinline]x \rightarrow a^-[/mathjaxinline] </td></tr><tr><td style="text-align:center; border:none">
or </td></tr><tr><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow \mathbf{a^-}} f(x) = L}.[/mathjaxinline] </td></tr></table>
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<h2 class="hd hd-2 unit-title">6. A few more limits</h2>
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Another function
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<p>
Let's explore the right and left hand limits of a few more functions. In this problem, we'll examine the function </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000013" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]g(x) = \frac{x}{\tan (2x)} \quad \textrm{as} \quad x\rightarrow 0^{\pm }.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
Here is a table of values of [mathjaxinline]g(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches 0 from the right: </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:left; border:none">
[mathjaxinline]x[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]g(x)[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:left; border:none">
1.0 </td>
<td style="text-align:left; border:none">
-0.458 </td>
</tr>
<tr>
<td style="text-align:left; border:none">
0.5 </td>
<td style="text-align:left; border:none">
0.321</td>
</tr>
<tr>
<td style="text-align:left; border:none">
0.1 </td>
<td style="text-align:left; border:none">
0.493 </td>
</tr>
<tr>
<td style="text-align:left; border:none">
0.05 </td>
<td style="text-align:left; border:none">
0.498 </td>
</tr>
<tr>
<td style="text-align:left; border:none">
0.01 </td>
<td style="text-align:left; border:none">
0.4999 </td>
</tr>
</table>
<p>
These data suggest that [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^+} g(x)} = 0.5.[/mathjaxinline] </p>
<p>
Use the calculator button below to find the <span style="color:#27408C"><b class="bf">left-hand</b></span> limit. <span style="color:#27408C"><b class="bf">This calculator is in radians!</b></span> </p>
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<text> As [mathjaxinline]x\rightarrow 0^-[/mathjaxinline], [mathjaxinline]g(x)[/mathjaxinline] gets closer and closer to a particular number [mathjaxinline]L[/mathjaxinline] ([mathjaxinline]g(x) \rightarrow L[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^-[/mathjaxinline], [mathjaxinline]g(x)[/mathjaxinline] gets bigger and bigger without bound ([mathjaxinline]g(x) \rightarrow +\infty[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^-[/mathjaxinline], [mathjaxinline]g(x)[/mathjaxinline] gets bigger and bigger in the negative direction without bound ([mathjaxinline]g(x) \rightarrow -\infty[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^-[/mathjaxinline], [mathjaxinline]g(x)[/mathjaxinline] approaches neither a finite number [mathjaxinline]L[/mathjaxinline], nor [mathjaxinline]+\infty[/mathjaxinline], nor [mathjaxinline]-\infty[/mathjaxinline]</text>
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What value does [mathjaxinline]g(x)[/mathjaxinline] get closer to as [mathjaxinline]x\rightarrow 0^-[/mathjaxinline]? </p>
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(If it approaches a finite number, enter the number below; in any other case, enter capital [mathjaxinline]DNE[/mathjaxinline] for "does not exist".) </p>
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Yet another function
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In this problem, we'll examine the function </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000014" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]h(x) = \frac{|x| + \sin x}{x^2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
as [mathjaxinline]x \rightarrow 0^\pm[/mathjaxinline]. </p>
<p>
Here is a table of values of [mathjaxinline]h(x)[/mathjaxinline] for values of [mathjaxinline]x[/mathjaxinline] that are close to zero on the left: </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:left; border:none">
[mathjaxinline]x[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]h(x)[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:left; border:none">
-1.0 </td>
<td style="text-align:left; border:none">
0.159 </td>
</tr>
<tr>
<td style="text-align:left; border:none">
-0.5 </td>
<td style="text-align:left; border:none">
0.082</td>
</tr>
<tr>
<td style="text-align:left; border:none">
-0.1 </td>
<td style="text-align:left; border:none">
0.017 </td>
</tr>
<tr>
<td style="text-align:left; border:none">
-0.01 </td>
<td style="text-align:left; border:none">
0.002 </td>
</tr>
<tr>
<td style="text-align:left; border:none">
-0.001 </td>
<td style="text-align:left; border:none">
0.0002 </td>
</tr>
</table>
<p>
These data suggest that [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^-} h(x)} = 0.[/mathjaxinline] </p>
<p>
Use a calculator to find the right-hand limit. <span style="color:#27408C"><b class="bf">Make sure your calculator is in radians!</b></span> </p>
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<text> As [mathjaxinline]x\rightarrow 0^+[/mathjaxinline], [mathjaxinline]h(x)[/mathjaxinline] gets closer and closer to a particular number [mathjaxinline]L[/mathjaxinline] ([mathjaxinline]h(x) \rightarrow L[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^+[/mathjaxinline], [mathjaxinline]h(x)[/mathjaxinline] gets bigger and bigger without bound ([mathjaxinline]h(x) \rightarrow +\infty[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^+[/mathjaxinline], [mathjaxinline]h(x)[/mathjaxinline] gets bigger and bigger in the negative direction without bound ([mathjaxinline]h(x) \rightarrow -\infty[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^+[/mathjaxinline], [mathjaxinline]h(x)[/mathjaxinline] approaches neither a finite number [mathjaxinline]L[/mathjaxinline], nor [mathjaxinline]+\infty[/mathjaxinline], nor [mathjaxinline]-\infty[/mathjaxinline]</text>
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What value does [mathjaxinline]h(x)[/mathjaxinline] get closer to as [mathjaxinline]x\rightarrow 0^+[/mathjaxinline]? </p>
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In this problem, we'll examine the function [mathjaxinline]\ j(x) = \sin (13/x)[/mathjaxinline], as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]0[/mathjaxinline] from the right. Use a calculator to figure out what [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^+} j(x)}[/mathjaxinline] might be. <span style="color:#27408C"><b class="bf">Make sure your calculator is in radians!</b></span> </p>
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<text> As [mathjaxinline]x\rightarrow 0^+[/mathjaxinline], [mathjaxinline]j(x)[/mathjaxinline] gets closer and closer to a particular number [mathjaxinline]L[/mathjaxinline] ([mathjaxinline]j(x) \rightarrow L[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^+[/mathjaxinline], [mathjaxinline]j(x)[/mathjaxinline] gets bigger and bigger without bound ([mathjaxinline]j(x) \rightarrow +\infty[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^+[/mathjaxinline], [mathjaxinline]j(x)[/mathjaxinline] gets bigger and bigger in the negative direction without bound ([mathjaxinline]j(x) \rightarrow -\infty[/mathjaxinline])</text>
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<text> As [mathjaxinline]x\rightarrow 0^+[/mathjaxinline], [mathjaxinline]j(x)[/mathjaxinline] approaches neither a finite number [mathjaxinline]L[/mathjaxinline], nor [mathjaxinline]+\infty[/mathjaxinline], nor [mathjaxinline]-\infty[/mathjaxinline]</text>
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What value does [mathjaxinline]j(x)[/mathjaxinline] get closer to as [mathjaxinline]x\rightarrow 0^+[/mathjaxinline]? </p>
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(If it approaches a finite number, enter the number below; in any other case, enter capital [mathjaxinline]DNE[/mathjaxinline] for "does not exist".) </p>
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<h2 class="hd hd-2 unit-title">7. Possible limit behaviors</h2>
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<p>
There are many possible limit behaviors. </p><ul class="itemize"><li><p>
The right-hand and left-hand limits may both exist and be equal. </p></li><li><p>
The right-hand and left-hand limits may both exist, but may fail to be equal. </p></li><li><p>
A right- and/or left-hand limit could fail to exist due to blowing up to [mathjaxinline]\pm \infty[/mathjaxinline]. (Example: Consider the function [mathjaxinline]1/x[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline].) In this case, we either say the limit blows up to infinity. We also say that the limit does not exist because [mathjaxinline]\infty[/mathjaxinline] is not a real number! </p></li><li><p>
A right- and/or left-hand limit could fail to exist because it oscillates between many values and never settles down. In this case we say the limit does not exist. </p></li></ul>
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<h2 class="hd hd-2 unit-title">8. Quick limit questions</h2>
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Left vs. right
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Suppose [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x)}[/mathjaxinline] exists and equals [mathjaxinline]R[/mathjaxinline]. Must [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x)}[/mathjaxinline] exist? </p>
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Suppose that [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x) = R}[/mathjaxinline] and [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x) = L}[/mathjaxinline]. Must [mathjaxinline]R = L[/mathjaxinline]? </p>
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Limit vs. function
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Suppose that [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x)}[/mathjaxinline] is some number [mathjaxinline]R[/mathjaxinline]. Must [mathjaxinline]\ f(a) = R[/mathjaxinline]? </p>
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Function vs. limit
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Suppose that [mathjaxinline]f(a) = K[/mathjaxinline]. Must [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x)} = K[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">9. The overall limit</h2>
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<h3 class="hd hd-2">The overall limit</h3>
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<h2 class="hd hd-2 unit-title">10. Limit definition</h2>
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<p><b class="bfseries">The Limit in Words</b></p><p>
If a function [mathjaxinline]f(x)[/mathjaxinline] approaches some value [mathjaxinline]L[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from <em>both the right and the left</em>, then <span style="color:#27408C"><b class="bf">the limit</b></span> of [mathjaxinline]f(x)[/mathjaxinline] exists and equals [mathjaxinline]L[/mathjaxinline]. </p><p><b class="bfseries">The Limit in Symbols</b></p><p>
If </p><table id="a0000000016" 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^+} f(x)} = \displaystyle {\lim _{x\rightarrow a^-} f(x)} = L[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
then </p><table id="a0000000017" 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} f(x) = L}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Alternatively, </p><table id="a0000000018" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f(x) \rightarrow L \quad \mathrm{as} \quad x\rightarrow a.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Remember that [mathjaxinline]x[/mathjaxinline] is approaching [mathjaxinline]a[/mathjaxinline] but does not equal [mathjaxinline]a[/mathjaxinline]. </p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Formal definition of limit<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
Formally, the statement [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}[/mathjaxinline] is defined as: </p><p><br/></p><p>
For all [mathjaxinline]\varepsilon >0,[/mathjaxinline] there exists some [mathjaxinline]\delta > 0[/mathjaxinline] such that if [mathjaxinline]0 < |x-a| < \delta ,[/mathjaxinline] then [mathjaxinline]|f(x)-L| < \varepsilon .[/mathjaxinline] </p><p><br/></p><p>
As is traditional, we use the Greek letters [mathjaxinline]\varepsilon[/mathjaxinline] and [mathjaxinline]\delta[/mathjaxinline]. </p><p>
Here is how one might understand that statement. The distance between two numbers [mathjaxinline]y[/mathjaxinline] and [mathjaxinline]z[/mathjaxinline] is given by [mathjaxinline]|y-z|[/mathjaxinline]. Thus, the very last part of the definition is saying that the distance from [mathjaxinline]f(x)[/mathjaxinline] to [mathjaxinline]L[/mathjaxinline] is less than [mathjaxinline]\varepsilon[/mathjaxinline]; one should think of [mathjaxinline]\varepsilon[/mathjaxinline] as representing a small distance. This close distance occurs if [mathjaxinline]0<|x-a|<\delta[/mathjaxinline]; that is, if [mathjaxinline]x[/mathjaxinline] is within some distance [mathjaxinline]\delta[/mathjaxinline] from [mathjaxinline]a[/mathjaxinline], but not necessarily if that distance is 0 (we don't care about [mathjaxinline]x = a[/mathjaxinline] itself). </p><p>
The "for all" and "there exists" clauses have to do with how small these distances need to get. We want [mathjaxinline]f(x)[/mathjaxinline] to eventually get arbitrarily close to [mathjaxinline]L[/mathjaxinline], so this statement needs to be satisfied no matter how small [mathjaxinline]\varepsilon[/mathjaxinline] gets. Given any choice of [mathjaxinline]\varepsilon[/mathjaxinline], we can satisfy the condition [mathjaxinline]|f(x) - L | < \varepsilon[/mathjaxinline] as long as [mathjaxinline]x[/mathjaxinline] gets close enough to [mathjaxinline]a[/mathjaxinline]; the proximity required is measured by [mathjaxinline]\delta[/mathjaxinline]. </p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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<h2 class="hd hd-2 unit-title">11. Limits from graphs</h2>
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Estimate limits
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Determine the following. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow (-2)^-} f(x)}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow (-2)^+} f(x)}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow (-2)} f(x)}=[/mathjaxinline]</p>
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Estimate limits 2
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Determine the following. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow 1^-} f(x)}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow 1} f(x)}=[/mathjaxinline]</p>
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Estimate limits 3
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Determine the following. </p>
<p>
(Type DNE if the value does not exist.) </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow 3^-} f(x)}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow 3^+} f(x)}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow 3} f(x)}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]f(3)=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">12. Review problems</h2>
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Function vs. limit 2
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True or false: If we know [mathjaxinline]f(a)[/mathjaxinline] exists, this means that [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x)}[/mathjaxinline] exists. </p>
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Double-sided limit
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Suppose that [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x)} = \displaystyle {\lim _{x\rightarrow a^+} f(x)} = 3[/mathjaxinline]. Which of the following must be true? </p>
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Floor function
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Recall that the floor function [mathjaxinline]\lfloor x \rfloor[/mathjaxinline] denotes the greatest integer less than or equal to [mathjaxinline]x[/mathjaxinline]. </p>
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(Calculate the following values, or enter DNE if a value does not exist.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow 2^-} \lfloor x \rfloor }=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow 2^+} \lfloor x \rfloor }=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow 2} \lfloor x \rfloor }=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\lfloor 2 \rfloor =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">13. Limit laws</h2>
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<h2 class="hd hd-2 unit-title">14. Limit Laws</h2>
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<p>
Suppose [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}, \qquad \displaystyle {\lim _{x\rightarrow a} g(x) = M}.[/mathjaxinline] </p><p>
Then we get the following Limit Laws: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
Limit Law for Addition: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)+g(x)\right] }[/mathjaxinline] </td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L+M[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
Limit Law for Subtraction: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)-g(x)\right] }[/mathjaxinline]</td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L-M[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
Limit Law for Multiplication: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)\cdot g(x)\right] }[/mathjaxinline]</td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L\cdot M.[/mathjaxinline] </td></tr></table><p>
We also have part of the Limit Law for Division: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
Limit Law for Division, Part 1: </td><td style="text-align:center; border:none">
If [mathjaxinline]M\ne 0[/mathjaxinline], then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)} } = \frac{L}{M}.[/mathjaxinline] </td></tr></table><p>
We will discuss what happens when [mathjaxinline]M=0[/mathjaxinline] in a later section! </p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Justifying the Limit Law for Multiplication<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
If [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}[/mathjaxinline] and [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} g(x) = M},[/mathjaxinline] then we can write </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:right; border:none">
[mathjaxinline]f(x)[/mathjaxinline]</td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L + \varepsilon _1[/mathjaxinline] </td></tr><tr><td style="text-align:right; border:none">
[mathjaxinline]g(x)[/mathjaxinline]</td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]M + \varepsilon _2[/mathjaxinline], </td></tr></table><p>
where [mathjaxinline]\varepsilon _1, \varepsilon _2 \rightarrow 0[/mathjaxinline] as [mathjaxinline]x \rightarrow a[/mathjaxinline]. Then </p><table id="a0000000019" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f(x)g(x) = LM + \varepsilon _1 M + \varepsilon _2 L + \varepsilon _1 \varepsilon _2.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
As [mathjaxinline]L[/mathjaxinline] and [mathjaxinline]M[/mathjaxinline] are constants and [mathjaxinline]\varepsilon _1, \varepsilon _2[/mathjaxinline] tend to zero, all three error terms [mathjaxinline]\varepsilon _1 M[/mathjaxinline], [mathjaxinline]\varepsilon _2 L[/mathjaxinline], and [mathjaxinline]\varepsilon _1 \varepsilon _2[/mathjaxinline] will go to zero as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline]. Hence </p><table id="a0000000020" 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} \left[f(x)\cdot g(x)\right] } = L\cdot M.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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<h2 class="hd hd-2 unit-title">15. Summary</h2>
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<p><b class="bfseries">Definitions of right-hand and left-hand limits</b></p><center><img src="/assets/courseware/v1/c0083509b7babe8df8e6e4d68de08c37/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim1_leftright.svg" width="400px" alt="See text below." style="margin: 10px 25px 25px 25px"/></center><p>
Suppose [mathjaxinline]\ f(x)[/mathjaxinline] gets really close to [mathjaxinline]R[/mathjaxinline] for values of [mathjaxinline]x[/mathjaxinline] that get really close to (but are not equal to) [mathjaxinline]a[/mathjaxinline] from the right. Then we say [mathjaxinline]R[/mathjaxinline] is the <span style="color:#27408C"><b class="bf">right-hand limit</b></span> of the function [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from the right. </p><p>
We write </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:center; border:none">
[mathjaxinline]f(x) \rightarrow R[/mathjaxinline] as [mathjaxinline]x \rightarrow a^+[/mathjaxinline] </td></tr><tr><td style="text-align:center; border:none">
or </td></tr><tr><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow \mathbf{a^+}} f(x) = R}[/mathjaxinline]. </td></tr></table><p>
If [mathjaxinline]\ f(x)[/mathjaxinline] gets really close to [mathjaxinline]L[/mathjaxinline] for values of [mathjaxinline]x[/mathjaxinline] that get really close to (but are not equal to) [mathjaxinline]a[/mathjaxinline] from the left, we say that [mathjaxinline]L[/mathjaxinline] is the <span style="color:#27408C"><b class="bf">left-hand limit</b></span> of the function [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from the left. </p><p>
We write </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:center; border:none">
[mathjaxinline]f(x) \rightarrow L[/mathjaxinline] as [mathjaxinline]x \rightarrow a^-[/mathjaxinline] </td></tr><tr><td style="text-align:center; border:none">
or </td></tr><tr><td style="text-align:center; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow \mathbf{a^-}} f(x) = L}.[/mathjaxinline] </td></tr></table><p><b class="bfseries">Possible limit behaviors</b></p><p>
There are many possible limit behaviors. </p><ul class="itemize"><li><p>
The right-hand and left-hand limits may both exist and be equal. </p></li><li><p>
The right-hand and left-hand limits may both exist, but may fail to be equal. </p></li><li><p>
A right- and/or left-hand limit could fail to exist due to blowing up to [mathjaxinline]\pm \infty[/mathjaxinline]. (Example: Consider the function [mathjaxinline]1/x[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline].) In this case, we either say the limit blows up to infinity. We also say that the limit does not exist because [mathjaxinline]\infty[/mathjaxinline] is not a real number! </p></li><li><p>
A right- and/or left-hand limit could fail to exist because it oscillates between many values and never settles down. In this case we say the limit does not exist. </p></li></ul><p><b class="bfseries">Definition of the Limit</b></p><p><b class="bfseries">The Limit in Words</b></p><p>
If a function [mathjaxinline]f(x)[/mathjaxinline] approaches some value [mathjaxinline]L[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline] from <em>both the right and the left</em>, then <span style="color:#27408C"><b class="bf">the limit</b></span> of [mathjaxinline]f(x)[/mathjaxinline] exists and equals [mathjaxinline]L[/mathjaxinline]. </p><p><b class="bfseries">The Limit in Symbols</b></p><p>
If </p><table id="a0000000021" 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^+} f(x)} = \displaystyle {\lim _{x\rightarrow a^-} f(x)} = L[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
then </p><table id="a0000000022" 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} f(x) = L}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Alternatively, </p><table id="a0000000023" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f(x) \rightarrow L \quad \mathrm{as} \quad x\rightarrow a.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Remember that [mathjaxinline]x[/mathjaxinline] is approaching [mathjaxinline]a[/mathjaxinline] but does not equal [mathjaxinline]a[/mathjaxinline]. </p><p><b class="bfseries">The Limit Laws:</b></p><p>
Suppose [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}, \qquad \displaystyle {\lim _{x\rightarrow a} g(x) = M}.[/mathjaxinline] </p><p>
Then we get the following Limit Laws: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
Limit Law for Addition: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)+g(x)\right] }[/mathjaxinline] </td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L+M[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
Limit Law for Subtraction: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)-g(x)\right] }[/mathjaxinline]</td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L-M[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
Limit Law for Multiplication: </td><td style="text-align:right; border:none">
[mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \left[f(x)\cdot g(x)\right] }[/mathjaxinline]</td><td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]L\cdot M.[/mathjaxinline] </td></tr></table><p>
We also have part of the Limit Law for Division: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
Limit Law for Division, Part 1: </td><td style="text-align:center; border:none">
If [mathjaxinline]M\ne 0[/mathjaxinline], then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)} } = \frac{L}{M}.[/mathjaxinline] </td></tr></table><p>
We will discuss what happens when [mathjaxinline]M=0[/mathjaxinline] in a later section! </p>
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