Difference between revisions of "009A Sample Midterm 2, Problem 1"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
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||We begin by noticing that we plug in <math>x=2</math> into
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{\sqrt{x^2+12}-4}{x-2},</math>
 
|-
 
|-
−
|
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|we get <math>\frac{0}{0}.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
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|Now, we multiply the numerator and denominator by the conjugate of the numerator.
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|-
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|Hence, we have
 
|-
 
|-
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|
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\lim _{x\rightarrow 2} \frac{\sqrt{x^2+12}-4}{x-2}} & = & \displaystyle{\lim_{x\rightarrow 2} \frac{(\sqrt{x^2+12}-4)}{(x-2)}\frac{(\sqrt{x^2+12}+4)}{(\sqrt{x^2+12}+4)}}\\
 +
&&\\
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& = & \displaystyle{\lim_{x\rightarrow 2} \frac{(x^2+12)-16}{(x-2)(\sqrt{x^2+12}+4)}}\\
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&&\\
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& = & \displaystyle{\lim_{x\rightarrow 2} \frac{x^2-4}{(x-2)(\sqrt{x^2+12}+4)}}\\
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&&\\
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& = & \displaystyle{\lim_{x\rightarrow 2} \frac{(x-2)(x+2)}{(x-2)(\sqrt{x^2+12}+4)}}\\
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&&\\
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& = & \displaystyle{\lim_{x\rightarrow 2} \frac{x+2}{\sqrt{x^2+12}+4}}\\
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&&\\
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& = & \displaystyle{\frac{4}{8}}\\
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&&\\
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& = & \displaystyle{\frac{1}{2}.}
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\end{array}</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|'''(a)'''  
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|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>\frac{1}{2}</math>
 
|-
 
|-
 
|'''(b)'''
 
|'''(b)'''

Revision as of 10:31, 17 February 2017

Evaluate the following limits.

a) Find 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 2} \frac{\sqrt{x^2+12}-4}{x-2}}
b) Find 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(3x)}{\sin(7x)} }
c) Evaluate 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 (\frac{\pi}{2})^-} \tan(x) }


Foundations:  
1. lim sinx/x
2. Left and right hand limit


Solution:

(a)

Step 1:  
We begin by noticing that we plug in 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=2} into
        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{\sqrt{x^2+12}-4}{x-2},}
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 \frac{0}{0}.}
Step 2:  
Now, we multiply the numerator and denominator by the conjugate of the numerator.
Hence, 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 2} \frac{\sqrt{x^2+12}-4}{x-2}} & = & \displaystyle{\lim_{x\rightarrow 2} \frac{(\sqrt{x^2+12}-4)}{(x-2)}\frac{(\sqrt{x^2+12}+4)}{(\sqrt{x^2+12}+4)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 2} \frac{(x^2+12)-16}{(x-2)(\sqrt{x^2+12}+4)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 2} \frac{x^2-4}{(x-2)(\sqrt{x^2+12}+4)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 2} \frac{(x-2)(x+2)}{(x-2)(\sqrt{x^2+12}+4)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 2} \frac{x+2}{\sqrt{x^2+12}+4}}\\ &&\\ & = & \displaystyle{\frac{4}{8}}\\ &&\\ & = & \displaystyle{\frac{1}{2}.} \end{array}}

(b)

Step 1:  
Step 2:  

(c)

Step 1:  
Step 2:  


Final Answer:  
    (a)     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{1}{2}}
(b)
(c)

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