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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Continuity</h3>
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<h2 class="hd hd-2 unit-title">2. How do we compute 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>
Identify points of <span style="color:#27408C"><b class="bf">discontinuity</b></span>. </p></li><li><p>
Recognize <span style="color:#27408C"><b class="bf">continuous and discontinuous functions</b></span> graphically and from formulas. </p></li><li><p>
Use <span style="color:#27408C"><b class="bf">limit laws</b></span> to find limits of sums and products of continuous functions. </p></li><li><p>
Apply the <span style="color:#27408C"><b class="bf">Intermediate Value Theorem</b></span>. </p></li></ul><p><b class="bfseries">Contents: 14 pages</b></p><p>
6 videos (23 minutes 1x speed) 17 questions </p>
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<h2 class="hd hd-2 unit-title">3. Continuity</h2>
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Limit versus function
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Suppose that [mathjaxinline]\ f(a) = 3[/mathjaxinline]. What can we say about [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x)}[/mathjaxinline]? </p>
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<h3 class="hd hd-2">Continuity at a point</h3>
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<p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if </p><table id="a0000000024" 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) } = f(a).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In particular, if either [mathjaxinline]\ f(a)[/mathjaxinline] or [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) }[/mathjaxinline] fails to exist, then [mathjaxinline]\ f[/mathjaxinline] is discontinuous at [mathjaxinline]a[/mathjaxinline]. </p><p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">right-continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x) } = f(a)[/mathjaxinline]. </p><p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">left-continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x) } = f(a)[/mathjaxinline]. </p><p>
It is sometimes useful to classify certain types of discontinuities. </p><p>
If the left-hand limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x) }[/mathjaxinline] and the right-hand limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x) }[/mathjaxinline] both exist at a point [mathjaxinline]x=a[/mathjaxinline], but they are not equal, then we say that [mathjaxinline]\ f[/mathjaxinline] has a <span style="color:#27408C"><b class="bf">jump discontinuity</b></span> at [mathjaxinline]x=a[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/4c60a6f5760133fa9aad6bea60c814ac/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim2_jump.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
If the overall limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) }[/mathjaxinline] exists (i.e., the left- and right-hand limits agree), but the overall limit does not equal [mathjaxinline]\ f(a)[/mathjaxinline], then we say that [mathjaxinline]\ f[/mathjaxinline] has a <span style="color:#27408C"><b class="bf">removable discontinuity</b></span> at [mathjaxinline]x=a[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/3367cd90f4550e53c7950aaaf93ff088/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim2_removable.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center>
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<h2 class="hd hd-2 unit-title">4. Continuity questions</h2>
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Continuity questions
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<p>
Here is the graph of a function [mathjaxinline]\ f[/mathjaxinline]. </p>
<center>
<img src="/assets/courseware/v1/33167aedcd9f8d843ac407158975e2ba/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim2_continuitytest2.svg" style="margin: 5px 5px 5px 5px; border:0px" width="400 px"/>
</center>
<p>
For each value of [mathjaxinline]a[/mathjaxinline], determine whether [mathjaxinline]\ f[/mathjaxinline] is continuous, right-continuous, or left-continuous at [mathjaxinline]x=a[/mathjaxinline]. </p>
<p>
<p style="display:inline">At [mathjaxinline]x=-2[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="inputtype option-input inline">
<select name="input_lim_2-tab4-problem1_2_1" id="input_lim_2-tab4-problem1_2_1" aria-describedby="status_lim_2-tab4-problem1_2_1">
<option value="option_lim_2-tab4-problem1_2_1_dummy_default">Select an option</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is continuous"> [mathjaxinline]\ f[/mathjaxinline] is continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous"> [mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous</option>
</select>
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<span class="status unanswered" id="status_lim_2-tab4-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p class="answer" id="answer_lim_2-tab4-problem1_2_1"/>
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<br/>
</p>
<p>
<p style="display:inline">At [mathjaxinline]x=-1[/mathjaxinline] </p>
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<select name="input_lim_2-tab4-problem1_3_1" id="input_lim_2-tab4-problem1_3_1" aria-describedby="status_lim_2-tab4-problem1_3_1">
<option value="option_lim_2-tab4-problem1_3_1_dummy_default">Select an option</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is continuous"> [mathjaxinline]\ f[/mathjaxinline] is continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous"> [mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous</option>
</select>
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<span class="status unanswered" id="status_lim_2-tab4-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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<p class="answer" id="answer_lim_2-tab4-problem1_3_1"/>
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<br/>
</p>
<p>
<p style="display:inline">At [mathjaxinline]x=1[/mathjaxinline] </p>
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<select name="input_lim_2-tab4-problem1_4_1" id="input_lim_2-tab4-problem1_4_1" aria-describedby="status_lim_2-tab4-problem1_4_1">
<option value="option_lim_2-tab4-problem1_4_1_dummy_default">Select an option</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is continuous"> [mathjaxinline]\ f[/mathjaxinline] is continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous"> [mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous</option>
</select>
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<span class="status unanswered" id="status_lim_2-tab4-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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<p class="answer" id="answer_lim_2-tab4-problem1_4_1"/>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">At [mathjaxinline]x=3[/mathjaxinline] </p>
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<select name="input_lim_2-tab4-problem1_5_1" id="input_lim_2-tab4-problem1_5_1" aria-describedby="status_lim_2-tab4-problem1_5_1">
<option value="option_lim_2-tab4-problem1_5_1_dummy_default">Select an option</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is continuous"> [mathjaxinline]\ f[/mathjaxinline] is continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous"> [mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous</option>
</select>
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<span class="status unanswered" id="status_lim_2-tab4-problem1_5_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_2-tab4-problem1_5_1"/>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">At [mathjaxinline]x=4[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div class="inputtype option-input inline">
<select name="input_lim_2-tab4-problem1_6_1" id="input_lim_2-tab4-problem1_6_1" aria-describedby="status_lim_2-tab4-problem1_6_1">
<option value="option_lim_2-tab4-problem1_6_1_dummy_default">Select an option</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is continuous"> [mathjaxinline]\ f[/mathjaxinline] is continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is right-continuous but not left-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous"> [mathjaxinline]\ f[/mathjaxinline] is left-continuous but not right-continuous</option>
<option value="[mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous"> [mathjaxinline]\ f[/mathjaxinline] is neither right- nor left-continuous</option>
</select>
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<span class="status unanswered" id="status_lim_2-tab4-problem1_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<span id="solution_lim_2-tab4-problem1_solution_1"/>
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Types of discontinuity
</h3>
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<img src="/assets/courseware/v1/33167aedcd9f8d843ac407158975e2ba/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim2_continuitytest2.svg" style="margin: 5px 5px 5px 5px; border:0px" width="400 px"/>
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<p>
List all [mathjaxinline]x[/mathjaxinline] values where there is a removable discontinuity. </p>
<p>
(Use decimals only. Separate multiple points with a comma; e.g. 1,2,3. If there aren't any, type "none".) </p>
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<p>
List all [mathjaxinline]x[/mathjaxinline] values where there is a jump discontinuity. </p>
<p>
(Use decimals only. Separate multiple points with a comma; e.g. 1,2,3. If there aren't any, type "none".) </p>
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<h2 class="hd hd-2 unit-title">5. More continuity questions</h2>
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Constant function
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Consider the constant function [mathjaxinline]\ f(x) = 3[/mathjaxinline]. At which values of [mathjaxinline]x[/mathjaxinline] is this function continuous? You may want to graph the function. </p>
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<text> All points [mathjaxinline]x[/mathjaxinline]</text>
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<input type="radio" name="input_lim_2-tab5-problem1_2_1" id="input_lim_2-tab5-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_2-tab5-problem1_2_1-choice_2-label" for="input_lim_2-tab5-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem1_2_1">
<text> All points except [mathjaxinline]x=0[/mathjaxinline]</text>
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<text> All points except [mathjaxinline]x=17[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_lim_2-tab5-problem1_2_1" id="input_lim_2-tab5-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_2-tab5-problem1_2_1-choice_4-label" for="input_lim_2-tab5-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem1_2_1">
<text> Nowhere</text>
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aria-labelledby="lim_2-tab5-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem2/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="lim_2-tab5-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem2-problem-progress" tabindex="-1">
Rational function 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
(Note that a rational function is the quotient of two polynomial functions.) </p>
<p>
Consider the function [mathjaxinline]g(x) = \frac{1}{x-17}[/mathjaxinline]. At which values of [mathjaxinline]x[/mathjaxinline] is this function continuous? You may want to graph the function. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_2-tab5-problem2_2_1">
<fieldset aria-describedby="status_lim_2-tab5-problem2_2_1">
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem2_2_1" id="input_lim_2-tab5-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_2-tab5-problem2_2_1-choice_1-label" for="input_lim_2-tab5-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem2_2_1">
<text> All points [mathjaxinline]x[/mathjaxinline]</text>
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<div class="field">
<input type="radio" name="input_lim_2-tab5-problem2_2_1" id="input_lim_2-tab5-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_2-tab5-problem2_2_1-choice_2-label" for="input_lim_2-tab5-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem2_2_1">
<text> All points except [mathjaxinline]x=0[/mathjaxinline]</text>
</label>
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<div class="field">
<input type="radio" name="input_lim_2-tab5-problem2_2_1" id="input_lim_2-tab5-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_2-tab5-problem2_2_1-choice_3-label" for="input_lim_2-tab5-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem2_2_1">
<text> All points except [mathjaxinline]x=17[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem2_2_1" id="input_lim_2-tab5-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_2-tab5-problem2_2_1-choice_4-label" for="input_lim_2-tab5-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem2_2_1">
<text> Nowhere</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem2_2_1" id="input_lim_2-tab5-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lim_2-tab5-problem2_2_1-choice_5-label" for="input_lim_2-tab5-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem2_2_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_lim_2-tab5-problem2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_2-tab5-problem2_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>
<div class="solution-span">
<span id="solution_lim_2-tab5-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Rational function 1" />
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<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lim_2-tab5-problem2" >
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data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem3" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem3/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="lim_2-tab5-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem3-problem-progress" tabindex="-1">
Rational function 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider the function [mathjaxinline]h(x) = \frac{1}{x^2+1}[/mathjaxinline]. At which values of [mathjaxinline]x[/mathjaxinline] is this function continuous? You may want to graph the function. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_2-tab5-problem3_2_1">
<fieldset aria-describedby="status_lim_2-tab5-problem3_2_1">
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem3_2_1" id="input_lim_2-tab5-problem3_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_2-tab5-problem3_2_1-choice_1-label" for="input_lim_2-tab5-problem3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem3_2_1">
<text> All points [mathjaxinline]x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem3_2_1" id="input_lim_2-tab5-problem3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_2-tab5-problem3_2_1-choice_2-label" for="input_lim_2-tab5-problem3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem3_2_1">
<text> All points except [mathjaxinline]x=0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem3_2_1" id="input_lim_2-tab5-problem3_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_2-tab5-problem3_2_1-choice_3-label" for="input_lim_2-tab5-problem3_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem3_2_1">
<text> All points except [mathjaxinline]x=17[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem3_2_1" id="input_lim_2-tab5-problem3_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_2-tab5-problem3_2_1-choice_4-label" for="input_lim_2-tab5-problem3_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem3_2_1">
<text> Nowhere</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem3_2_1" id="input_lim_2-tab5-problem3_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lim_2-tab5-problem3_2_1-choice_5-label" for="input_lim_2-tab5-problem3_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem3_2_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_lim_2-tab5-problem3_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_2-tab5-problem3_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>
<div class="solution-span">
<span id="solution_lim_2-tab5-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Rational function 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_2-tab5-problem3" >
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<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c68510f0ea6d11efb8100e0e3c45b88f" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-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">
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<div id="problem_lim_2-tab5-problem4" class="problems-wrapper" role="group"
aria-labelledby="lim_2-tab5-problem4-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem4" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem4/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
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<h3 class="hd hd-3 problem-header" id="lim_2-tab5-problem4-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem4-problem-progress" tabindex="-1">
Absolute value
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@lim_2-tab5-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider the function [mathjaxinline]j(x) = |x|[/mathjaxinline]. At which values of [mathjaxinline]x[/mathjaxinline] is this function continuous? You may want to graph the function. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lim_2-tab5-problem4_2_1">
<fieldset aria-describedby="status_lim_2-tab5-problem4_2_1">
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem4_2_1" id="input_lim_2-tab5-problem4_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lim_2-tab5-problem4_2_1-choice_1-label" for="input_lim_2-tab5-problem4_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem4_2_1">
<text> All points [mathjaxinline]x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem4_2_1" id="input_lim_2-tab5-problem4_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lim_2-tab5-problem4_2_1-choice_2-label" for="input_lim_2-tab5-problem4_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem4_2_1">
<text> All points except [mathjaxinline]x=0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem4_2_1" id="input_lim_2-tab5-problem4_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lim_2-tab5-problem4_2_1-choice_3-label" for="input_lim_2-tab5-problem4_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem4_2_1">
<text> All points except [mathjaxinline]x=17[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem4_2_1" id="input_lim_2-tab5-problem4_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lim_2-tab5-problem4_2_1-choice_4-label" for="input_lim_2-tab5-problem4_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem4_2_1">
<text> Nowhere</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lim_2-tab5-problem4_2_1" id="input_lim_2-tab5-problem4_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lim_2-tab5-problem4_2_1-choice_5-label" for="input_lim_2-tab5-problem4_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lim_2-tab5-problem4_2_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_lim_2-tab5-problem4_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lim_2-tab5-problem4_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>
<div class="solution-span">
<span id="solution_lim_2-tab5-problem4_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Absolute value" />
<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_2-tab5-problem4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lim_2-tab5-problem4">
<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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<h2 class="hd hd-2 unit-title">6. Overall continuity</h2>
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<p>
A function [mathjaxinline]\ f(x)[/mathjaxinline] is <span style="color:#27408C"><b class="bf">continuous</b></span> if for every point [mathjaxinline]c[/mathjaxinline] in the domain of [mathjaxinline]\ f(x)[/mathjaxinline], the function [mathjaxinline]\ f[/mathjaxinline] is continuous at the point [mathjaxinline]x=c[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">7. Continuity continued</h2>
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True or false: [mathjaxinline]\displaystyle {\lim _{x\rightarrow 3} \lfloor x \rfloor } = \lfloor 3 \rfloor[/mathjaxinline]. </p>
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(Recall that [mathjaxinline]\lfloor x \rfloor[/mathjaxinline] denotes the greatest integer less than or equal to [mathjaxinline]x[/mathjaxinline].) </p>
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Reciprocal function
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<h2 class="hd hd-2 unit-title">8. Limit Laws and continuity</h2>
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Sum of continuous functions
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Let [mathjaxinline]\ f(x) =x^2 + x[/mathjaxinline]. What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 3} f(x)}[/mathjaxinline]? (Enter DNE if the limit does not exist.) </p>
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Is [mathjaxinline]\ f[/mathjaxinline] continuous at [mathjaxinline]x = 1776[/mathjaxinline]? </p>
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Quotients of continuous functions
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Let [mathjaxinline]g(x) = \frac{x}{\cos x}[/mathjaxinline]. Is [mathjaxinline]g[/mathjaxinline] continuous at [mathjaxinline]x = 0[/mathjaxinline]? </p>
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Is [mathjaxinline]g[/mathjaxinline] continuous for all [mathjaxinline]x[/mathjaxinline]? </p>
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<h3 class="hd hd-2">Limit Laws and Continuity</h3>
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<h2 class="hd hd-2 unit-title">9. Review of continuity</h2>
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Identify continuous functions
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Which of the following functions are continuous at all [mathjaxinline]x[/mathjaxinline]? </p>
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(Select as many options as will work.) </p>
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If [mathjaxinline]p(x)[/mathjaxinline] is a polynomial, then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} p(x) = p(a)}[/mathjaxinline] for all values of [mathjaxinline]a[/mathjaxinline]. </p>
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If [mathjaxinline]q(x) = 17x - 3\sin x[/mathjaxinline], then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} q(x) = q(a)}[/mathjaxinline] for all values of [mathjaxinline]a[/mathjaxinline]. </p>
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<p>
If [mathjaxinline]r(x) = x^2 \lfloor x \rfloor + 3[/mathjaxinline], then [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} r(x) = r(a)}[/mathjaxinline] for all values of [mathjaxinline]a[/mathjaxinline]. </p>
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True or False 4
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<p>
If [mathjaxinline]\ f[/mathjaxinline] has a discontinuity at [mathjaxinline]x=3[/mathjaxinline], and [mathjaxinline]g[/mathjaxinline] also has a discontinuity at [mathjaxinline]x=3[/mathjaxinline], then the function [mathjaxinline]h(x) = f(x)+g(x)[/mathjaxinline] must also have a discontinuity at [mathjaxinline]x=3[/mathjaxinline]. </p>
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<option value="True"> True</option>
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<h2 class="hd hd-2 unit-title">10. Catalog of continuous functions</h2>
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<p>
Note: we have not proven all of the following facts, but you should feel free to use them. </p><p>
The following functions are continuous at <em>all real numbers</em>: </p><ul class="itemize"><li><p>
all polynomials </p></li><li><p>
[mathjaxinline]\sqrt [3]{x}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]|x|[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\cos x[/mathjaxinline] and [mathjaxinline]\sin x[/mathjaxinline] </p></li><li><p>
exponential functions [mathjaxinline]a^ x[/mathjaxinline] with base [mathjaxinline]a>0[/mathjaxinline] </p></li></ul><p>
The following functions are continuous (or right continuous) <em>at the specified values</em> of [mathjaxinline]x[/mathjaxinline]: </p><ul class="itemize"><li><p>
[mathjaxinline]\sqrt {x}[/mathjaxinline], for [mathjaxinline]x \geq 0[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\tan x[/mathjaxinline], at all [mathjaxinline]x[/mathjaxinline] where it is defined </p></li><li><p>
logarithmic functions [mathjaxinline]\log _{a} x[/mathjaxinline] with base [mathjaxinline]a>0[/mathjaxinline], for [mathjaxinline]x > 0[/mathjaxinline] </p></li></ul>
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<h2 class="hd hd-2 unit-title">11. IVT Intro</h2>
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Let's start off with a quick question. Suppose that [mathjaxinline]\ f(1) = 2[/mathjaxinline] and [mathjaxinline]\ f(4) = 4[/mathjaxinline]. Must the graph of [mathjaxinline]\ f[/mathjaxinline] intersect the line [mathjaxinline]y = 1[/mathjaxinline]? </p>
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Suppose that [mathjaxinline]\ f(1) = 2[/mathjaxinline] and [mathjaxinline]\ f(4) = 4[/mathjaxinline]. Must the graph of [mathjaxinline]\ f[/mathjaxinline] intersect the line [mathjaxinline]y = 3[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">12. Intermediate Value Theorem</h2>
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If [mathjaxinline]\ f[/mathjaxinline] is a function which is continuous on the interval [mathjaxinline][a,b][/mathjaxinline], and [mathjaxinline]M[/mathjaxinline] lies between the values of [mathjaxinline]\ f(a)[/mathjaxinline] and [mathjaxinline]\ f(b)[/mathjaxinline], then there is at least one point [mathjaxinline]c[/mathjaxinline] between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] such that [mathjaxinline]\ f(c) = M.[/mathjaxinline] </p><p>
(A function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">continuous on a closed interval</b></span> [mathjaxinline][a,b][/mathjaxinline] if it is right-continuous at [mathjaxinline]a[/mathjaxinline], left-continuous at [mathjaxinline]b[/mathjaxinline], and continuous at all points between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline].) </p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">What makes the Intermediate Value Theorem a deep result?<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
The Intermediate Value Theorem is profound because it takes information that is local by nature, and allows you to conclude a global result. Continuity at a point is local information, because it only requires knowledge of the function's behavior near that point. But somehow, if we know this fact at every point on an interval, then the Intermediate Value Theorem tells us something about the overall, or global behavior - namely, that the function has to take on a particular value, or its graph has to cross a certain line. </p><p>
This is intimately tied in with the properties of the real number system. We could have the same definition of continuity for functions if we only worked with, say, the rational numbers (numbers which can be written as a fraction of integers), but the Intermediate Value Theorem would not work if that were the case. It is only when we move to the real numbers that the Intermediate Value Theorem holds true. </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">13. Basics</h2>
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IVT 0
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On June 1, Trixie owned 2 rabbits. On August 31, she was the proud owner of 4 rabbits. Must there have been some time at which she owned exactly 3.3 rabbits? </p>
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<h3 class="hd hd-2">Using the IVT</h3>
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<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_lim_2-tab13-video1">Downloads and transcripts</h3>
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IVT 1
</h3>
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<div class="problem">
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<p>
Let [mathjaxinline]\ f(x) = \sqrt {x} - \cos x[/mathjaxinline]. According to the Intermediate Value Theorem, must there be some [mathjaxinline]c[/mathjaxinline] between [mathjaxinline]x=0[/mathjaxinline] and [mathjaxinline]x=9[/mathjaxinline] where [mathjaxinline]\ f(c) = 0[/mathjaxinline]? </p>
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IVT 2
</h3>
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<div class="problem">
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<p>
Let [mathjaxinline]g(x) = 4x^{-3} - x - \sin (\pi x)[/mathjaxinline]. Does the Intermediate Value Theorem imply that there must be some [mathjaxinline]c[/mathjaxinline] between [mathjaxinline]x=-1[/mathjaxinline] and [mathjaxinline]x=1[/mathjaxinline] where [mathjaxinline]g(c) = 0[/mathjaxinline]? </p>
<p>
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<h2 class="hd hd-2 unit-title">14. Roots</h2>
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IVT 3
</h3>
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<div class="problem">
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<p>
A polynomial [mathjaxinline]p(x)[/mathjaxinline] takes values </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]p(-2)[/mathjaxinline] </td>
<td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]6[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]p(-1)[/mathjaxinline] </td>
<td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]-2[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]p(0)[/mathjaxinline] </td>
<td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]-3[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]p(1)[/mathjaxinline] </td>
<td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]1[/mathjaxinline] </td>
</tr>
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]p(2)[/mathjaxinline] </td>
<td style="text-align:center; border:none">
[mathjaxinline]=[/mathjaxinline] </td>
<td style="text-align:left; border:none">
[mathjaxinline]6[/mathjaxinline]. </td>
</tr>
</table>
<p>
We also know that [mathjaxinline]p(x) &gt; 6[/mathjaxinline] if [mathjaxinline]x &gt; 2[/mathjaxinline] or if [mathjaxinline]x &lt; -2[/mathjaxinline]. What is the strongest statement that you can conclude from this data? </p>
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<h2 class="hd hd-2 unit-title">15. Summary</h2>
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<p><b class="bfseries">Definition of continuity at a point</b></p><p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if </p><table id="a0000000028" 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) } = f(a).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In particular, if either [mathjaxinline]\ f(a)[/mathjaxinline] or [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) }[/mathjaxinline] fails to exist, then [mathjaxinline]\ f[/mathjaxinline] is discontinuous at [mathjaxinline]a[/mathjaxinline]. </p><p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">right-continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x) } = f(a)[/mathjaxinline]. </p><p>
We say that a function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">left-continuous at a point</b></span> [mathjaxinline]x = a[/mathjaxinline] if [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x) } = f(a)[/mathjaxinline]. </p><p><b class="bfseries">Types of Discontinuities</b></p><p>
It is sometimes useful to classify certain types of discontinuities. </p><p>
If the left-hand limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^-} f(x) }[/mathjaxinline] and the right-hand limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a^+} f(x) }[/mathjaxinline] both exist at a point [mathjaxinline]x=a[/mathjaxinline], but they are not equal, then we say that [mathjaxinline]\ f[/mathjaxinline] has a <span style="color:#27408C"><b class="bf">jump discontinuity</b></span> at [mathjaxinline]x=a[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/4c60a6f5760133fa9aad6bea60c814ac/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim2_jump.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p>
If the overall limit [mathjaxinline]\displaystyle {\lim _{x\rightarrow a} f(x) }[/mathjaxinline] exists (i.e., the left- and right-hand limits agree), but the overall limit does not equal [mathjaxinline]\ f(a)[/mathjaxinline], then we say that [mathjaxinline]\ f[/mathjaxinline] has a <span style="color:#27408C"><b class="bf">removable discontinuity</b></span> at [mathjaxinline]x=a[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/3367cd90f4550e53c7950aaaf93ff088/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u0lim2_removable.svg" width="400 px" style="margin: 5px 5px 5px 5px; border:0px"/></center><p><b class="bfseries">Definition of continuous functions</b></p><p>
A function [mathjaxinline]\ f(x)[/mathjaxinline] is <span style="color:#27408C"><b class="bf">continuous</b></span> if for every point [mathjaxinline]c[/mathjaxinline] in the domain of [mathjaxinline]\ f(x)[/mathjaxinline], the function [mathjaxinline]\ f[/mathjaxinline] is continuous at the point [mathjaxinline]x=c[/mathjaxinline]. </p><p><b class="bfseries">Basic Continuous Functions</b></p><p>
Note: we have not proven all of the following facts, but you should feel free to use them. </p><p>
The following functions are continuous at <em>all real numbers</em>: </p><ul class="itemize"><li><p>
all polynomials </p></li><li><p>
[mathjaxinline]\sqrt [3]{x}[/mathjaxinline] </p></li><li><p>
[mathjaxinline]|x|[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\cos x[/mathjaxinline] and [mathjaxinline]\sin x[/mathjaxinline] </p></li><li><p>
exponential functions [mathjaxinline]a^ x[/mathjaxinline] with base [mathjaxinline]a>0[/mathjaxinline] </p></li></ul><p>
The following functions are continuous <em>at the specified values</em> of [mathjaxinline]x[/mathjaxinline]: </p><ul class="itemize"><li><p>
[mathjaxinline]\sqrt {x}[/mathjaxinline], for [mathjaxinline]x >0[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\tan x[/mathjaxinline], at all [mathjaxinline]x[/mathjaxinline] where it is defined </p></li><li><p>
logarithmic functions [mathjaxinline]\log _{a} x[/mathjaxinline] with base [mathjaxinline]a>0[/mathjaxinline], for [mathjaxinline]x > 0[/mathjaxinline] </p></li></ul><p><b class="bfseries">Limit laws and continuity</b></p><p>
If the functions [mathjaxinline]\ f[/mathjaxinline] and [mathjaxinline]\ g[/mathjaxinline] are continuous everywhere, then: </p><ul class="itemize"><li><p>
[mathjaxinline]\ f+g[/mathjaxinline] is continuous everywhere. </p></li><li><p>
[mathjaxinline]\ f-g[/mathjaxinline] is continuous everywhere. </p></li><li><p>
[mathjaxinline]\ f\cdot g[/mathjaxinline] is continuous everywhere. </p></li><li><p>
[mathjaxinline]\displaystyle \ \frac{f}{g}[/mathjaxinline] is continuous where it is defined. </p></li></ul><p><b class="bfseries">Intermediate Value Theorem</b></p><p>
If [mathjaxinline]\ f[/mathjaxinline] is a function which is continuous on the interval [mathjaxinline][a,b][/mathjaxinline], and [mathjaxinline]M[/mathjaxinline] lies between the values of [mathjaxinline]\ f(a)[/mathjaxinline] and [mathjaxinline]\ f(b)[/mathjaxinline], then there is at least one point [mathjaxinline]c[/mathjaxinline] between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] such that [mathjaxinline]\ f(c) = M.[/mathjaxinline] </p><p>
(A function [mathjaxinline]\ f[/mathjaxinline] is <span style="color:#27408C"><b class="bf">continuous on a closed interval</b></span> [mathjaxinline][a,b][/mathjaxinline] if it is right-continuous at [mathjaxinline]a[/mathjaxinline], left-continuous at [mathjaxinline]b[/mathjaxinline], and continuous at all points between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline].) </p>
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