Difference between revisions of "009A Sample Final 3, Problem 1"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
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|Since &nbsp;<math style="vertical-align: -12px">\lim_{x\rightarrow 8} 3 =3\ne 0,</math>
 +
|-
 +
|we have
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 +
\displaystyle{-2} & = & \displaystyle{\lim _{x\rightarrow 8} \bigg[\frac{xf(x)}{3}\bigg]}\\
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&&\\
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& = & \displaystyle{\frac{\lim_{x\rightarrow 8} xf(x)}{\lim_{x\rightarrow 8} 3}}\\
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&&\\
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& = & \displaystyle{\frac{\lim_{x\rightarrow 8} xf(x)}{3}.}
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\end{array}</math>
 
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|-
 
|
 
|
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
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|If we multiply both sides of the last equation by &nbsp;<math>3,</math>&nbsp; we get
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>-6=\lim_{x\rightarrow 8} xf(x)).</math>
 +
|-
 +
|Now, using properties of limits, we have
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{10} & = & \displaystyle{\bigg(\lim_{x\rightarrow 8} x\bigg)\bigg(\lim_{x\rightarrow 8}f(x)\bigg)}\\
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&&\\
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& = & \displaystyle{8\lim_{x\rightarrow 8} f(x).}\\
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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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|Solving for &nbsp;<math style="vertical-align: -12px">\lim_{x\rightarrow 8} f(x)</math>&nbsp; in the last equation,
 +
|-
 +
|we get
 
|-
 
|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp; <math> \lim_{x\rightarrow 8} f(x)=\frac{-3}{4}.</math>
 
|}
 
|}
  
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|'''(a)'''
 
|'''(a)'''
 
|-
 
|-
−
|'''(b)'''
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|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; <math>\frac{-3}{4}</math>
 
|-
 
|-
 
|'''(c)'''
 
|'''(c)'''
 
|}
 
|}
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 20:24, 6 March 2017

Find each of the following limits if it exists. If you think the limit does not exist provide a reason.

(a)  

(b)  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 8} f(x),}   given that  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 8}\frac{xf(x)}{3}=-2}

(c)  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 -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}}


Foundations:  
1. 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} g(x)\neq 0,}   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 a} \frac{f(x)}{g(x)}=\frac{\displaystyle{\lim_{x\rightarrow a} f(x)}}{\displaystyle{\lim_{x\rightarrow a} g(x)}}.}
2.  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 0} \frac{\sin x}{x}=1}


Solution:

(a)

Step 1:  
Step 2:  

(b)

Step 1:  
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 8} 3 =3\ne 0,}
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{-2} & = & \displaystyle{\lim _{x\rightarrow 8} \bigg[\frac{xf(x)}{3}\bigg]}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 8} xf(x)}{\lim_{x\rightarrow 8} 3}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 8} xf(x)}{3}.} \end{array}}
Step 2:  
If we multiply both sides of the last equation by  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 3,}   we get
        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 -6=\lim_{x\rightarrow 8} xf(x)).}
Now, using properties of limits, 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{10} & = & \displaystyle{\bigg(\lim_{x\rightarrow 8} x\bigg)\bigg(\lim_{x\rightarrow 8}f(x)\bigg)}\\ &&\\ & = & \displaystyle{8\lim_{x\rightarrow 8} f(x).}\\ \end{array}}
Step 3:  
Solving for  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 8} f(x)}   in the last equation,
we get

        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 8} f(x)=\frac{-3}{4}.}

(c)

Step 1:  
Step 2:  


Final Answer:  
(a)
   (b)    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 \frac{-3}{4}}
(c)

Return to Sample Exam

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