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<h2 class="hd hd-2 unit-title">1. Limits of quotients</h2>
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<h2 class="hd hd-2 unit-title">2. How do we compute limits of quotients?</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>
Distinguish the three cases of the Division Limit Law. </p></li><li><p>
Compute limits of quotients of functions. </p></li><li><p>
Determine when a limit is [mathjaxinline]\pm \infty[/mathjaxinline]. </p></li></ul><p><b class="bfseries">Contents: 15 pages</b></p><p>
6 videos (22 minutes 1x speed) 18 questions </p>
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<h2 class="hd hd-2 unit-title">3. Limits and division</h2>
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In the video we looked at [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0} \frac{2x}{x}}[/mathjaxinline], where the numerator and denominator are both approaching zero. How can a quotient such as this have a limit that exists? Let's explore. </p>
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When taking a limit as [mathjaxinline]x \rightarrow 0[/mathjaxinline], do we consider what happens when [mathjaxinline]x=0[/mathjaxinline]? </p>
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As [mathjaxinline]x \rightarrow 0[/mathjaxinline], will the numerator [mathjaxinline]2x[/mathjaxinline] be exactly zero? </p>
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As [mathjaxinline]x \rightarrow 0[/mathjaxinline], will the denominator [mathjaxinline]x[/mathjaxinline] be exactly zero? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="inputtype option-input ">
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<h2 class="hd hd-2 unit-title">4. Quotients of small numbers</h2>
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Let's consider the general case of a limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)}}[/mathjaxinline] where both the numerator and denominator approach 0. </p><p>
When [mathjaxinline]x[/mathjaxinline] is near [mathjaxinline]a[/mathjaxinline], both [mathjaxinline]\ f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] are close to zero but – crucially – not necessarily equal to zero. So, to evaluate [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)}}[/mathjaxinline], we need to figure out what happens when we divide one small number by another. We're going to ask you to actually divide some really small numbers by some really small numbers to find out for yourself! </p>
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Before we get to numerical evidence, we'd like you to make a guess as to what the answer is. If all we know is that [mathjaxinline]\ f(x)[/mathjaxinline] is small (say, between zero and 0.01) and that [mathjaxinline]g(x)[/mathjaxinline] is small (again, between zero and 0.01), what do you think will definitely be true about the quotient [mathjaxinline]\frac{f(x)}{g(x)}[/mathjaxinline]? Don't worry about getting this wrong - all answers will be accepted. We just want you to come up with a hypothesis! </p>
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<text> I think it will be really big</text>
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<text> I think it will be a number that is neither big nor small</text>
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<text> I think it could be any of these; it depends!</text>
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Quotients of small numbers
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<p>
In this problem, we're going to figure out what [mathjaxinline]\displaystyle \frac{f(x)}{g(x)}[/mathjaxinline] might be close to if the numerator is really small and the denominator is really small. You might want a calculator! </p>
<p>
(For purposes of this question, "really big" means over 100, and "really close to zero" means less than [mathjaxinline]0.01[/mathjaxinline].) </p>
<p>
If [mathjaxinline]\ f(x) = 0.00000013[/mathjaxinline] and [mathjaxinline]g(x) = 0.0025[/mathjaxinline], then [mathjaxinline]\displaystyle \frac{f(x)}{g(x)}[/mathjaxinline] is...<br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input inline">
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<option value="Really close to zero"> Really close to zero</option>
<option value="Really big"> Really big</option>
<option value="Not big but not close to zero"> Not big but not close to zero</option>
<option value="Undefined"> Undefined</option>
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<p>
If [mathjaxinline]\ f(x) = 0.000033[/mathjaxinline] and [mathjaxinline]g(x) = 0.000029[/mathjaxinline], then [mathjaxinline]\displaystyle \frac{f(x)}{g(x)}[/mathjaxinline] is...<br/><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div class="inputtype option-input inline">
<select name="input_lim_3-tab4-problem2_3_1" id="input_lim_3-tab4-problem2_3_1" aria-describedby="status_lim_3-tab4-problem2_3_1">
<option value="option_lim_3-tab4-problem2_3_1_dummy_default">Select an option</option>
<option value="Really close to zero"> Really close to zero</option>
<option value="Really big"> Really big</option>
<option value="Not big but not close to zero"> Not big but not close to zero</option>
<option value="Undefined"> Undefined</option>
</select>
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<p class="answer" id="answer_lim_3-tab4-problem2_3_1"/>
</div></div><br/></p>
<p>
If [mathjaxinline]\ f(x) = 0.0094[/mathjaxinline] and [mathjaxinline]g(x) = 0.00000023[/mathjaxinline], then [mathjaxinline]\displaystyle \frac{f(x)}{g(x)}[/mathjaxinline] is...<br/><div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div class="inputtype option-input inline">
<select name="input_lim_3-tab4-problem2_4_1" id="input_lim_3-tab4-problem2_4_1" aria-describedby="status_lim_3-tab4-problem2_4_1">
<option value="option_lim_3-tab4-problem2_4_1_dummy_default">Select an option</option>
<option value="Really close to zero"> Really close to zero</option>
<option value="Really big"> Really big</option>
<option value="Not big but not close to zero"> Not big but not close to zero</option>
<option value="Undefined"> Undefined</option>
</select>
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<p>
If [mathjaxinline]\ f(x) = 0.00023[/mathjaxinline] and [mathjaxinline]g(x) = 0.000081[/mathjaxinline], then [mathjaxinline]\displaystyle \frac{f(x)}{g(x)}[/mathjaxinline] is...<br/><div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div class="inputtype option-input inline">
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<option value="option_lim_3-tab4-problem2_5_1_dummy_default">Select an option</option>
<option value="Really close to zero"> Really close to zero</option>
<option value="Really big"> Really big</option>
<option value="Not big but not close to zero"> Not big but not close to zero</option>
<option value="Undefined"> Undefined</option>
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<h3 class="hd hd-3 problem-header" id="lim_3-tab4-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab4-problem3-problem-progress" tabindex="-1">
Quotient of small numbers
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<p>
So now that we have some evidence in, let's answer the question for real. </p>
<p>
If all we know is that [mathjaxinline]\ f(x)[/mathjaxinline] is small (say, between zero and 0.01) and that [mathjaxinline]g(x)[/mathjaxinline] is small (again, between zero and 0.01), what can we definitively say about the quotient [mathjaxinline]\displaystyle \frac{f(x)}{g(x)}[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab4-problem3_2_1">
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<text> It will be really close to zero</text>
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<input type="radio" name="input_lim_3-tab4-problem3_2_1" id="input_lim_3-tab4-problem3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab4-problem3_2_1-choice_2-label" for="input_lim_3-tab4-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab4-problem3_2_1">
<text> It will be really big</text>
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<text> It will be a number that is neither big nor small</text>
</label>
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<text> It could be any of these - it depends!</text>
</label>
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<h2 class="hd hd-2 unit-title">6. Limit Law for Division</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@lim_3-tab6-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="6cdfa802ea7811efb8110e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@lim_3-tab6-text1" data-graded="False" 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">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
If [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}[/mathjaxinline] and [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} g(x) = M},[/mathjaxinline] then: </p><ul class="itemize"><li><p>
If [mathjaxinline]M \ne 0,[/mathjaxinline] then [mathjaxinline]\displaystyle { \lim _{x\rightarrow a} \frac{f(x)}{g(x)} = \frac{L}{M}}[/mathjaxinline]. </p></li><li><p>
If [mathjaxinline]M = 0[/mathjaxinline] but [mathjaxinline]L \ne 0,[/mathjaxinline] then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)}}[/mathjaxinline] does not exist. </p></li><li><p>
If both [mathjaxinline]M = 0[/mathjaxinline] and [mathjaxinline]L = 0,[/mathjaxinline] then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)}}[/mathjaxinline] might exist, or it might not exist. More work is necessary to determine whether the last type of limit exists, and what it is if it does exist. </p></li></ul>
</div>
</div>
</div>
</div>
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<h2 class="hd hd-2 unit-title">7. Limit Laws</h2>
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<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab7-problem1">
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aria-labelledby="lim_3-tab7-problem1-problem-title"
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<h3 class="hd hd-3 problem-header" id="lim_3-tab7-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab7-problem1-problem-progress" tabindex="-1">
Limit Laws
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab7-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Suppose that [mathjaxinline]\displaystyle {\lim _{x\rightarrow -1} f(x) = 0}[/mathjaxinline], [mathjaxinline]\displaystyle {\lim _{x\rightarrow -1} g(x) = 17}[/mathjaxinline], and [mathjaxinline]\displaystyle {\lim _{x\rightarrow -1} h(x) = 0}[/mathjaxinline]. </p>
<p>
Evaluate the following limits. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \lim _{x\rightarrow -1} g(x)h(x)[/mathjaxinline]</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab7-problem1_2_1">
<fieldset aria-describedby="status_lim_3-tab7-problem1_2_1">
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_2_1" id="input_lim_3-tab7-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab7-problem1_2_1-choice_1-label" for="input_lim_3-tab7-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_2_1">
<text> 0</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_2_1" id="input_lim_3-tab7-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab7-problem1_2_1-choice_2-label" for="input_lim_3-tab7-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_2_1">
<text> 1</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_2_1" id="input_lim_3-tab7-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab7-problem1_2_1-choice_3-label" for="input_lim_3-tab7-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_2_1">
<text> 17</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_2_1" id="input_lim_3-tab7-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab7-problem1_2_1-choice_4-label" for="input_lim_3-tab7-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_2_1">
<text> Something else</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_2_1" id="input_lim_3-tab7-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lim_3-tab7-problem1_2_1-choice_5-label" for="input_lim_3-tab7-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_2_1">
<text> Does not exist</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_2_1" id="input_lim_3-tab7-problem1_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lim_3-tab7-problem1_2_1-choice_6-label" for="input_lim_3-tab7-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_2_1">
<text> Cannot be determined based on the information given</text>
</label>
</div>
<span id="answer_lim_3-tab7-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_3-tab7-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \lim _{x\rightarrow -1} \frac{g(x)}{f(x)}[/mathjaxinline]</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab7-problem1_3_1">
<fieldset aria-describedby="status_lim_3-tab7-problem1_3_1">
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_3_1" id="input_lim_3-tab7-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab7-problem1_3_1-choice_1-label" for="input_lim_3-tab7-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_3_1">
<text> 0</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_3_1" id="input_lim_3-tab7-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab7-problem1_3_1-choice_2-label" for="input_lim_3-tab7-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_3_1">
<text> 1</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_3_1" id="input_lim_3-tab7-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab7-problem1_3_1-choice_3-label" for="input_lim_3-tab7-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_3_1">
<text> 17</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_3_1" id="input_lim_3-tab7-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab7-problem1_3_1-choice_4-label" for="input_lim_3-tab7-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_3_1">
<text> Something else</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_3_1" id="input_lim_3-tab7-problem1_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lim_3-tab7-problem1_3_1-choice_5-label" for="input_lim_3-tab7-problem1_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_3_1">
<text> Does not exist</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_3_1" id="input_lim_3-tab7-problem1_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lim_3-tab7-problem1_3_1-choice_6-label" for="input_lim_3-tab7-problem1_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_3_1">
<text> Cannot be determined based on the information given</text>
</label>
</div>
<span id="answer_lim_3-tab7-problem1_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_3-tab7-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \lim _{x\rightarrow -1} f(x) + g(x) + h(x)[/mathjaxinline]</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab7-problem1_4_1">
<fieldset aria-describedby="status_lim_3-tab7-problem1_4_1">
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_4_1" id="input_lim_3-tab7-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab7-problem1_4_1-choice_1-label" for="input_lim_3-tab7-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_4_1">
<text> 0</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_4_1" id="input_lim_3-tab7-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab7-problem1_4_1-choice_2-label" for="input_lim_3-tab7-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_4_1">
<text> 1</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_4_1" id="input_lim_3-tab7-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab7-problem1_4_1-choice_3-label" for="input_lim_3-tab7-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_4_1">
<text> 17</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_4_1" id="input_lim_3-tab7-problem1_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab7-problem1_4_1-choice_4-label" for="input_lim_3-tab7-problem1_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_4_1">
<text> Something else</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_4_1" id="input_lim_3-tab7-problem1_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lim_3-tab7-problem1_4_1-choice_5-label" for="input_lim_3-tab7-problem1_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_4_1">
<text> Does not exist</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_4_1" id="input_lim_3-tab7-problem1_4_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lim_3-tab7-problem1_4_1-choice_6-label" for="input_lim_3-tab7-problem1_4_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_4_1">
<text> Cannot be determined based on the information given</text>
</label>
</div>
<span id="answer_lim_3-tab7-problem1_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_3-tab7-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \lim _{x\rightarrow -1} \frac{f(x)}{h(x)}[/mathjaxinline]</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab7-problem1_5_1">
<fieldset aria-describedby="status_lim_3-tab7-problem1_5_1">
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_5_1" id="input_lim_3-tab7-problem1_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab7-problem1_5_1-choice_1-label" for="input_lim_3-tab7-problem1_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_5_1">
<text> 0</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_5_1" id="input_lim_3-tab7-problem1_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab7-problem1_5_1-choice_2-label" for="input_lim_3-tab7-problem1_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_5_1">
<text> 1</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_5_1" id="input_lim_3-tab7-problem1_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab7-problem1_5_1-choice_3-label" for="input_lim_3-tab7-problem1_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_5_1">
<text> 17</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_5_1" id="input_lim_3-tab7-problem1_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab7-problem1_5_1-choice_4-label" for="input_lim_3-tab7-problem1_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_5_1">
<text> Something else</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_5_1" id="input_lim_3-tab7-problem1_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lim_3-tab7-problem1_5_1-choice_5-label" for="input_lim_3-tab7-problem1_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_5_1">
<text> Does not exist</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_5_1" id="input_lim_3-tab7-problem1_5_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lim_3-tab7-problem1_5_1-choice_6-label" for="input_lim_3-tab7-problem1_5_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_5_1">
<text> Cannot be determined based on the information given</text>
</label>
</div>
<span id="answer_lim_3-tab7-problem1_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_3-tab7-problem1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \lim _{x\rightarrow -1} \frac{f(x)+h(x)}{g(x)}[/mathjaxinline]</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab7-problem1_6_1">
<fieldset aria-describedby="status_lim_3-tab7-problem1_6_1">
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_6_1" id="input_lim_3-tab7-problem1_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab7-problem1_6_1-choice_1-label" for="input_lim_3-tab7-problem1_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_6_1">
<text> 0</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab7-problem1_6_1" id="input_lim_3-tab7-problem1_6_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab7-problem1_6_1-choice_2-label" for="input_lim_3-tab7-problem1_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab7-problem1_6_1">
<text> 1</text>
</label>
</div>
<div class="field">
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<h2 class="hd hd-2 unit-title">8. Using the division limit law</h2>
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<h2 class="hd hd-2 unit-title">9. Division limit questions</h2>
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Calculate [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^+}} \frac{2\cos x+1}{x^2 + x}[/mathjaxinline]. </p>
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Calculate [mathjaxinline]\displaystyle {\lim _{x\rightarrow 2} \frac{1/x + x^2}{x-3}}[/mathjaxinline]. </p>
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Quotient limit 3
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What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 2} \frac{12/x - 3x}{2-3x + x^2} }[/mathjaxinline]? </p>
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(Enter DNE if the limit does not exist.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_lim_3-tab9-problem3_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_lim_3-tab9-problem3_2_1" id="input_lim_3-tab9-problem3_2_1" data-input-id="lim_3-tab9-problem3_2_1" value="" aria-describedby="status_lim_3-tab9-problem3_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_lim_3-tab9-problem3_2_1"/>
<span class="status unanswered" id="status_lim_3-tab9-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lim_3-tab9-problem3_2_1" class="answer"/>
<div id="input_lim_3-tab9-problem3_2_1_preview" class="equation">
\(\)
<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_lim_3-tab9-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Quotient limit 3" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lim_3-tab9-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lim_3-tab9-problem3">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lim_3-tab9-problem3-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lim_3-tab9-problem3-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lim_3-tab9-problem3-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
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</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6cdfa802ea7811efb8110e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem4" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lim_3-tab9-problem4" class="problems-wrapper" role="group"
aria-labelledby="lim_3-tab9-problem4-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem4" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem4/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lim_3-tab9-problem4-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem4-problem-progress" tabindex="-1">
Quotient limit 4
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 3} \frac{2x^2-10x + 12}{x^3-6x^2 +9x} }[/mathjaxinline]? </p>
<p>
(Enter DNE if the limit does not exist.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_lim_3-tab9-problem4_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_lim_3-tab9-problem4_2_1" id="input_lim_3-tab9-problem4_2_1" data-input-id="lim_3-tab9-problem4_2_1" value="" aria-describedby="status_lim_3-tab9-problem4_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_lim_3-tab9-problem4_2_1"/>
<span class="status unanswered" id="status_lim_3-tab9-problem4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lim_3-tab9-problem4_2_1" class="answer"/>
<div id="input_lim_3-tab9-problem4_2_1_preview" class="equation">
\(\)
<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_lim_3-tab9-problem4_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Quotient limit 4" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lim_3-tab9-problem4" >
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</button>
<div class="submission-feedback" id="submission_feedback_lim_3-tab9-problem4">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
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</div>
</div>
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<span class="icon fa fa-save" aria-hidden="true"></span>
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<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
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<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
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"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem5">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6cdfa802ea7811efb8110e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem5" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lim_3-tab9-problem5" class="problems-wrapper" role="group"
aria-labelledby="lim_3-tab9-problem5-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem5" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem5/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lim_3-tab9-problem5-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem5-problem-progress" tabindex="-1">
Quotient limit 5
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab9-problem5-problem-progress"></div>
<div class="problem">
<div>
<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0} \frac{3x^3 + x^2}{x^3 + x^2 + x} }[/mathjaxinline]? </p>
<p>
(Enter DNE if the limit does not exist.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_lim_3-tab9-problem5_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_lim_3-tab9-problem5_2_1" id="input_lim_3-tab9-problem5_2_1" data-input-id="lim_3-tab9-problem5_2_1" value="" aria-describedby="status_lim_3-tab9-problem5_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_lim_3-tab9-problem5_2_1"/>
<span class="status unanswered" id="status_lim_3-tab9-problem5_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lim_3-tab9-problem5_2_1" class="answer"/>
<div id="input_lim_3-tab9-problem5_2_1_preview" class="equation">
\(\)
<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_lim_3-tab9-problem5_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Quotient limit 5" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lim_3-tab9-problem5" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lim_3-tab9-problem5">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
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is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
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<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
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<div class="notification warning notification-save
is-hidden"
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<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lim_3-tab9-problem5-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
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tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lim_3-tab9-problem5-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
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"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6cdfa802ea7811efb8110e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@lim_3-tab10" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">10. Review problems</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6cdfa802ea7811efb8110e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem1" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lim_3-tab10-problem1" class="problems-wrapper" role="group"
aria-labelledby="lim_3-tab10-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lim_3-tab10-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem1-problem-progress" tabindex="-1">
Basic limit stuff
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
True or false: In general, if [mathjaxinline]\ f[/mathjaxinline] is a function, then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = f(a)}[/mathjaxinline]. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
<select name="input_lim_3-tab10-problem1_2_1" id="input_lim_3-tab10-problem1_2_1" aria-describedby="status_lim_3-tab10-problem1_2_1">
<option value="option_lim_3-tab10-problem1_2_1_dummy_default">Select an option</option>
<option value="True"> True</option>
<option value="False"> False</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_3-tab10-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_lim_3-tab10-problem1_2_1"/>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lim_3-tab10-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>
<div class="action">
<input type="hidden" name="problem_id" value="Basic limit stuff" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lim_3-tab10-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lim_3-tab10-problem1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
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is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lim_3-tab10-problem1-problem-title">
</span>
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<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
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<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lim_3-tab10-problem1-problem-title">None
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"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6cdfa802ea7811efb8110e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem2" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
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<div id="problem_lim_3-tab10-problem2" class="problems-wrapper" role="group"
aria-labelledby="lim_3-tab10-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lim_3-tab10-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem2-problem-progress" tabindex="-1">
Basic limit stuff 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
True or false: If [mathjaxinline]\ f(x) = \sin (x^3)[/mathjaxinline], then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = f(a)}[/mathjaxinline] for all [mathjaxinline]a[/mathjaxinline]. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input ">
<select name="input_lim_3-tab10-problem2_2_1" id="input_lim_3-tab10-problem2_2_1" aria-describedby="status_lim_3-tab10-problem2_2_1">
<option value="option_lim_3-tab10-problem2_2_1_dummy_default">Select an option</option>
<option value="True"> True</option>
<option value="False"> False</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_3-tab10-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_lim_3-tab10-problem2_2_1"/>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lim_3-tab10-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Basic limit stuff 2" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lim_3-tab10-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lim_3-tab10-problem2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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aria-labelledby="lim_3-tab10-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem3" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem3/handler/xmodule_handler"
data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="lim_3-tab10-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem3-problem-progress" tabindex="-1">
More limits 1
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<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0} \sqrt {\cos x + 3} }[/mathjaxinline]? </p>
<p>
(Enter DNE if the limit does not exist.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_lim_3-tab10-problem3_2_1" class="inputtype formulaequationinput">
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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">
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More limits 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 2} \frac{4-x^2}{2^ x - 3} }[/mathjaxinline]? </p>
<p>
(Enter DNE if the limit does not exist.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_lim_3-tab10-problem4_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
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<span class="trailing_text" id="trailing_text_lim_3-tab10-problem4_2_1"/>
<span class="status unanswered" id="status_lim_3-tab10-problem4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lim_3-tab10-problem4_2_1" class="answer"/>
<div id="input_lim_3-tab10-problem4_2_1_preview" class="equation">
\(\)
<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_lim_3-tab10-problem4_solution_1"/>
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More limits 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_3-tab10-problem5-problem-progress"></div>
<div class="problem">
<div>
<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow -1} \frac{2x^2+7x+5}{x+1} }[/mathjaxinline]? </p>
<p>
(Enter DNE if the limit does not exist.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_lim_3-tab10-problem5_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
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<span class="trailing_text" id="trailing_text_lim_3-tab10-problem5_2_1"/>
<span class="status unanswered" id="status_lim_3-tab10-problem5_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lim_3-tab10-problem5_2_1" class="answer"/>
<div id="input_lim_3-tab10-problem5_2_1_preview" class="equation">
\(\)
<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_lim_3-tab10-problem5_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="More limits 3" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lim_3-tab10-problem5" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lim_3-tab10-problem5">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
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data-graded="False">
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<span class="sr">Loading…</span>
</p>
</div>
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<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6cdfa802ea7811efb8110e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@lim_3-tab11" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">11. Limits that don't exist</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@lim_3-tab11-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="6cdfa802ea7811efb8110e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@lim_3-tab11-video1" data-graded="False" 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">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Infinite Limits</h3>
<div
id="video_lim_3-tab11-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@lim_3-tab11-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@lim_3-tab11-video1/handler/publish_completion", "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@lim_3-tab11-video1/handler/transcript/translation/__lang__", "prioritizeHls": false, "autoplay": false, "savedVideoPosition": 0.0, "streams": "1.00:IbSLV6xv7OQ", "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@lim_3-tab11-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">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="lim_3-tab11-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<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>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<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_lim_3-tab11-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_lim_3-tab11-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@lim_3-tab11-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<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@lim_3-tab11-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@lim_3-tab11-text1">
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In the video, we wrote that a limit was [mathjaxinline]+\infty[/mathjaxinline]. The [mathjaxinline]+\infty[/mathjaxinline] is nice because it emphasizes the difference between [mathjaxinline]+\infty[/mathjaxinline] and [mathjaxinline]-\infty[/mathjaxinline], but sometimes you will see people write that a limit is [mathjaxinline]\infty[/mathjaxinline] and not use the plus sign. It generally means the same thing as [mathjaxinline]+\infty[/mathjaxinline].
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The other side
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<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^-} }\frac{1}{x}[/mathjaxinline]? </p>
<p>
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<input type="radio" name="input_lim_3-tab11-problem1_2_1" id="input_lim_3-tab11-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab11-problem1_2_1-choice_1-label" for="input_lim_3-tab11-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab11-problem1_2_1">
<text> It is [mathjaxinline]+\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab11-problem1_2_1" id="input_lim_3-tab11-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab11-problem1_2_1-choice_2-label" for="input_lim_3-tab11-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab11-problem1_2_1">
<text> It is [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
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<input type="radio" name="input_lim_3-tab11-problem1_2_1" id="input_lim_3-tab11-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab11-problem1_2_1-choice_3-label" for="input_lim_3-tab11-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab11-problem1_2_1">
<text> The limit does not exist, and it is neither [mathjaxinline]+\infty[/mathjaxinline] nor [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab11-problem1_2_1" id="input_lim_3-tab11-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab11-problem1_2_1-choice_4-label" for="input_lim_3-tab11-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab11-problem1_2_1">
<text> It exists</text>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<span id="solution_lim_3-tab11-problem1_solution_1"/>
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Overall limit
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<p>
Given the results for [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^-} }\frac{1}{x}[/mathjaxinline] and [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^+} }\frac{1}{x}[/mathjaxinline], what should we say about [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0} }\frac{1}{x}[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab11-problem2_2_1">
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<input type="radio" name="input_lim_3-tab11-problem2_2_1" id="input_lim_3-tab11-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab11-problem2_2_1-choice_1-label" for="input_lim_3-tab11-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab11-problem2_2_1">
<text> It is [mathjaxinline]+\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab11-problem2_2_1" id="input_lim_3-tab11-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab11-problem2_2_1-choice_2-label" for="input_lim_3-tab11-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab11-problem2_2_1">
<text> It is [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab11-problem2_2_1" id="input_lim_3-tab11-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab11-problem2_2_1-choice_3-label" for="input_lim_3-tab11-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab11-problem2_2_1">
<text> The limit does not exist, and it is neither [mathjaxinline]+\infty[/mathjaxinline] nor [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab11-problem2_2_1" id="input_lim_3-tab11-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab11-problem2_2_1-choice_4-label" for="input_lim_3-tab11-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab11-problem2_2_1">
<text> It exists</text>
</label>
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<span class="status unanswered" id="status_lim_3-tab11-problem2_2_1" data-tooltip="Not yet answered.">
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<h2 class="hd hd-2 unit-title">12. Another function</h2>
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Another function
</h3>
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<div class="problem">
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<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^+} }\frac{1}{x^2}[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab12-problem1_2_1">
<fieldset aria-describedby="status_lim_3-tab12-problem1_2_1">
<div class="field">
<input type="radio" name="input_lim_3-tab12-problem1_2_1" id="input_lim_3-tab12-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab12-problem1_2_1-choice_1-label" for="input_lim_3-tab12-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab12-problem1_2_1">
<text> It is [mathjaxinline]+\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab12-problem1_2_1" id="input_lim_3-tab12-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab12-problem1_2_1-choice_2-label" for="input_lim_3-tab12-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab12-problem1_2_1">
<text> It is [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab12-problem1_2_1" id="input_lim_3-tab12-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab12-problem1_2_1-choice_3-label" for="input_lim_3-tab12-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab12-problem1_2_1">
<text> The limit does not exist, and it is neither [mathjaxinline]+\infty[/mathjaxinline] nor [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab12-problem1_2_1" id="input_lim_3-tab12-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab12-problem1_2_1-choice_4-label" for="input_lim_3-tab12-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab12-problem1_2_1">
<text> It exists</text>
</label>
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<span id="answer_lim_3-tab12-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_3-tab12-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^-} }\frac{1}{x^2}[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_3-tab12-problem1_3_1">
<fieldset aria-describedby="status_lim_3-tab12-problem1_3_1">
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<input type="radio" name="input_lim_3-tab12-problem1_3_1" id="input_lim_3-tab12-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_3-tab12-problem1_3_1-choice_1-label" for="input_lim_3-tab12-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab12-problem1_3_1">
<text> It is [mathjaxinline]+\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab12-problem1_3_1" id="input_lim_3-tab12-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab12-problem1_3_1-choice_2-label" for="input_lim_3-tab12-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab12-problem1_3_1">
<text> It is [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab12-problem1_3_1" id="input_lim_3-tab12-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab12-problem1_3_1-choice_3-label" for="input_lim_3-tab12-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab12-problem1_3_1">
<text> The limit does not exist, and it is neither [mathjaxinline]+\infty[/mathjaxinline] nor [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_3-tab12-problem1_3_1" id="input_lim_3-tab12-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab12-problem1_3_1-choice_4-label" for="input_lim_3-tab12-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab12-problem1_3_1">
<text> It exists</text>
</label>
</div>
<span id="answer_lim_3-tab12-problem1_3_1"/>
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<span class="status unanswered" id="status_lim_3-tab12-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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<span id="solution_lim_3-tab12-problem1_solution_1"/>
</div></p>
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Given the results for [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^-} }\frac{1}{x^2}[/mathjaxinline] and [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0^+} }\frac{1}{x^2}[/mathjaxinline], what should we say about [mathjaxinline]\displaystyle {\lim _{x\rightarrow 0} }\frac{1}{x^2}[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">13. Infinite limits 2</h2>
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<h2 class="hd hd-2 unit-title">14. What is the limit?</h2>
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What is the limit?
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What is [mathjaxinline]\displaystyle \lim _{x\rightarrow 0^+} \frac{2x-1}{1-\cos x}[/mathjaxinline] </p>
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<text> The limit does not exist, and it is neither [mathjaxinline]+\infty[/mathjaxinline] nor [mathjaxinline]-\infty[/mathjaxinline]</text>
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<p>
What is [mathjaxinline]\displaystyle \lim _{x\rightarrow 0^-} \frac{2x-1}{1-\cos x}[/mathjaxinline] </p>
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<text> It is [mathjaxinline]+\infty[/mathjaxinline]</text>
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<text> It is [mathjaxinline]-\infty[/mathjaxinline]</text>
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<text> The limit does not exist, and it is neither [mathjaxinline]+\infty[/mathjaxinline] nor [mathjaxinline]-\infty[/mathjaxinline]</text>
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<text> It exists</text>
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<h2 class="hd hd-2 unit-title">15. Division involving infinite limits</h2>
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Division involving infinite limits
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Suppose that [mathjaxinline]\ \displaystyle {\lim _{x\rightarrow 1} f(x) = -2}[/mathjaxinline] and [mathjaxinline]\ \displaystyle {\lim _{x\rightarrow 1^+} g(x) = +\infty }[/mathjaxinline]. </p>
<p>
What is [mathjaxinline]\displaystyle \lim _{x\rightarrow 1^+} \frac{f(x)}{g(x)}[/mathjaxinline]? </p>
<p>
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<text> 0</text>
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<input type="radio" name="input_lim_3-tab15-problem1_2_1" id="input_lim_3-tab15-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab15-problem1_2_1-choice_2-label" for="input_lim_3-tab15-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab15-problem1_2_1">
<text> -2</text>
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<text> [mathjaxinline]+\infty[/mathjaxinline]</text>
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<input type="radio" name="input_lim_3-tab15-problem1_2_1" id="input_lim_3-tab15-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab15-problem1_2_1-choice_4-label" for="input_lim_3-tab15-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab15-problem1_2_1">
<text> [mathjaxinline]-\infty[/mathjaxinline]</text>
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<input type="radio" name="input_lim_3-tab15-problem1_2_1" id="input_lim_3-tab15-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lim_3-tab15-problem1_2_1-choice_5-label" for="input_lim_3-tab15-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab15-problem1_2_1">
<text> Does not exist and is not [mathjaxinline]\pm \infty[/mathjaxinline]</text>
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<text> Something else</text>
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<text> Cannot be determined based on the information given</text>
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<p>
What is [mathjaxinline]\displaystyle \lim _{x\rightarrow 1^+} \frac{g(x)}{f(x)}[/mathjaxinline]? </p>
<p>
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<text> 0</text>
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<input type="radio" name="input_lim_3-tab15-problem1_3_1" id="input_lim_3-tab15-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_3-tab15-problem1_3_1-choice_2-label" for="input_lim_3-tab15-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab15-problem1_3_1">
<text> -2</text>
</label>
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<input type="radio" name="input_lim_3-tab15-problem1_3_1" id="input_lim_3-tab15-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_3-tab15-problem1_3_1-choice_3-label" for="input_lim_3-tab15-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab15-problem1_3_1">
<text> [mathjaxinline]+\infty[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_lim_3-tab15-problem1_3_1" id="input_lim_3-tab15-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_3-tab15-problem1_3_1-choice_4-label" for="input_lim_3-tab15-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab15-problem1_3_1">
<text> [mathjaxinline]-\infty[/mathjaxinline]</text>
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<text> Does not exist and is not [mathjaxinline]\pm \infty[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_lim_3-tab15-problem1_3_1" id="input_lim_3-tab15-problem1_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lim_3-tab15-problem1_3_1-choice_6-label" for="input_lim_3-tab15-problem1_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lim_3-tab15-problem1_3_1">
<text> Something else</text>
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<text> Cannot be determined based on the information given</text>
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<h2 class="hd hd-2 unit-title">16. Summary</h2>
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<p><b class="bfseries">Limit Law for Division</b></p><p>
If [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) = L}[/mathjaxinline] and [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} g(x) = M},[/mathjaxinline] then: </p><ul class="itemize"><li><p>
If [mathjaxinline]M \ne 0,[/mathjaxinline] then [mathjaxinline]\displaystyle { \lim _{x\rightarrow a} \frac{f(x)}{g(x)} = \frac{L}{M}}[/mathjaxinline]. </p></li><li><p>
If [mathjaxinline]M = 0[/mathjaxinline] but [mathjaxinline]L \ne 0,[/mathjaxinline] then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)}}[/mathjaxinline] does not exist. </p></li><li><p>
If both [mathjaxinline]M = 0[/mathjaxinline] and [mathjaxinline]L = 0,[/mathjaxinline] then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} \frac{f(x)}{g(x)}}[/mathjaxinline] might exist, or it might not exist. More work is necessary to determine whether the last type of limit exists, and what it is if it does exist. </p></li></ul>
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