Difference between revisions of "009A Sample Final 3, Problem 4"

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<span class="exam">If you think &nbsp;<math style="vertical-align: -4px">f</math>&nbsp; is not continuous at &nbsp;<math style="vertical-align: -4px">x=0,</math>&nbsp; what kind of discontinuity is it?
 
<span class="exam">If you think &nbsp;<math style="vertical-align: -4px">f</math>&nbsp; is not continuous at &nbsp;<math style="vertical-align: -4px">x=0,</math>&nbsp; what kind of discontinuity is it?
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[[009A Sample Final 3, Problem 4 Solution|'''<u>Solution</u>''']]
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Foundations: &nbsp;
 
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|-
 
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|<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous at &nbsp;<math style="vertical-align: 0px">x=a</math>&nbsp; if
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -14px">\lim_{x\rightarrow a^+}f(x)=\lim_{x\rightarrow a^-}f(x)=f(a).</math>
 
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|}
 
  
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[[009A Sample Final 3, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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'''Solution:'''
 
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 1: &nbsp;
 
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|-
 
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|We first calculate &nbsp;<math style="vertical-align: -14px">\lim_{x\rightarrow 0^+}f(x).</math>&nbsp; We have
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 
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\displaystyle{\lim_{x\rightarrow 0^+}f(x)} & = & \displaystyle{\lim_{x\rightarrow 0^+} x-\cos x}\\
 
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&&\\
 
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& = & \displaystyle{0-\cos(0)}\\
 
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&&\\
 
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& = & \displaystyle{-1.}
 
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\end{array}</math>
 
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
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|-
 
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|Now, we calculate &nbsp;<math style="vertical-align: -14px">\lim_{x\rightarrow 0^-}f(x).</math>&nbsp; We have
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 
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\displaystyle{\lim_{x\rightarrow 0^-}f(x)} & = & \displaystyle{\lim_{x\rightarrow 0^-} \frac{x}{|x|}}\\
 
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&&\\
 
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& = & \displaystyle{\lim_{x\rightarrow 0^-} \frac{x}{-x}}\\
 
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&&\\
 
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& = & \displaystyle{\lim_{x\rightarrow 0^-} -1}\\
 
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&&\\
 
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& = & \displaystyle{-1.}
 
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\end{array}</math>
 
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 3: &nbsp;
 
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|-
 
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|Since
 
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|
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -15px">\lim_{x\rightarrow 3^+}f(x)=\lim_{x\rightarrow 3^-}f(x)=-1,</math>
 
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|we have
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow 3} f(x)=-1.</math>
 
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|But,
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>f(0)=0\ne \lim_{x\rightarrow 3} f(x).</math>
 
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|Thus, <math style="vertical-align: -5px">f(x)</math>&nbsp; is not continuous.
 
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|It is a jump discontinuity.
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Final Answer: &nbsp;
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is not continuous. It is a jump discontinuity.
 
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|}
 
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 16:38, 2 December 2017

Discuss, without graphing, if the following function is continuous at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0.}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \left\{ \begin{array}{lr} \frac{x}{|x|} & \text{if }x < 0\\ 0 & \text{if }x = 0\\ x-\cos x & \text{if }x > 0 \end{array} \right. }

If you think  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}   is not continuous at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0,}   what kind of discontinuity is it?

Solution


Detailed Solution


Return to Sample Exam

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