Difference between revisions of "009A Sample Final 2, Problem 2"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
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|Based on the description of &nbsp;<math style="vertical-align: -5px">f(x),</math>&nbsp;
 +
|-
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|we know &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous on
 
|-
 
|-
−
|
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|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: -5px">(-\infty,3)\cup (3,\infty).</math>&nbsp;
 
|-
 
|-
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|
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|Now, we need to see if &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous at &nbsp;<math style="vertical-align: 0px">x=3.</math>&nbsp;
 
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|-
 
|
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
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|We have
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{\lim_{x\rightarrow 3^+}f(x)} & = & \displaystyle{\lim_{x\rightarrow 3^+} \frac{x^2-2x-3}{x-3}}\\
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&&\\
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& = & \displaystyle{\lim_{x\rightarrow 3^+} \frac{(x-3)(x+1)}{x-3}}\\
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&&\\
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& = & \displaystyle{\lim_{x\rightarrow 3^+} x+1}\\
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&&\\
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& = & \displaystyle{4.}
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\end{array}</math>
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|-
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|Similarly,
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\lim_{x\rightarrow 3^-}f(x)=4.</math>
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|-
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|Since
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math style="vertical-align: 0px">\lim_{x\rightarrow 3^-}f(x)=\lim_{x\rightarrow 3^+}f(x)</math>&nbsp;
 +
|-
 +
|we have
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\lim_{x\rightarrow 3}f(x)=4.</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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|But, since
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>f(3)=5,</math>
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|-
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|we have
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\lim_{x\rightarrow 3}f(x)\ne f(3).</math>
 +
|-
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|Therefore, &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is not continuous at &nbsp;<math style="vertical-align: 0px">x=3.</math>&nbsp;
 
|-
 
|-
−
|
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|&nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous only on &nbsp;<math style="vertical-align: -5px">(-\infty,3)\cup (3,\infty).</math>&nbsp;
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|
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|&nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous on &nbsp;<math style="vertical-align: -5px">(-\infty,3)\cup (3,\infty).</math>&nbsp;
 
|}
 
|}
 
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 18:08, 7 March 2017

Let

For what values of    is  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}   continuous?

Foundations:  
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)}   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=a}   if
       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 \lim_{x\rightarrow a^+}f(x)=\lim_{x\rightarrow a^-}f(x)=f(a).}


Solution:

Step 1:  
Based on the description of  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),}  
we know  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)}   is continuous on
       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 (-\infty,3)\cup (3,\infty).}  
Now, we need to see if  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)}   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=3.}  
Step 2:  
We have
       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 \begin{array}{rcl} \displaystyle{\lim_{x\rightarrow 3^+}f(x)} & = & \displaystyle{\lim_{x\rightarrow 3^+} \frac{x^2-2x-3}{x-3}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 3^+} \frac{(x-3)(x+1)}{x-3}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 3^+} x+1}\\ &&\\ & = & \displaystyle{4.} \end{array}}
Similarly,
       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 \lim_{x\rightarrow 3^-}f(x)=4.}
Since
       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 \lim_{x\rightarrow 3^-}f(x)=\lim_{x\rightarrow 3^+}f(x)}  
we have
       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 \lim_{x\rightarrow 3}f(x)=4.}
Step 3:  
But, since
       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(3)=5,}
we have
       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 \lim_{x\rightarrow 3}f(x)\ne f(3).}
Therefore,  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)}   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=3.}  
 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)}   is continuous only on  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 (-\infty,3)\cup (3,\infty).}  


Final Answer:  
 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)}   is continuous on  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 (-\infty,3)\cup (3,\infty).}  

Return to Sample Exam

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