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

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!Step 2:  
 
!Step 2:  
 
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!Step 1:    
 
!Step 1:    
 
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|First, we have
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}} & = & \displaystyle{\lim _{x\rightarrow \infty} \frac{(-2x^3-2x+3)}{(3x^3+3x^2-3)} \frac{(\frac{1}{x^3})}{(\frac{1}{x^3})}}\\
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&&\\
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& = & \displaystyle{\lim_{x\rightarrow 0} \frac{-2-\frac{2}{x^2}+\frac{3}{x^3}}{3+\frac{3}{x}-\frac{3}{x^3}}}.
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\end{array}</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Now, we use the properties of limits to get
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}} & = & \displaystyle{\lim_{x\rightarrow \infty} \frac{-2-\frac{2}{x^2}+\frac{3}{x^3}}{3+\frac{3}{x}-\frac{3}{x^3}}}\\
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&&\\
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& = & \displaystyle{\frac{\lim_{x\rightarrow \infty} (-2-\frac{2}{x^2}+\frac{3}{x^3})}{\lim_{x\rightarrow \infty} (3+\frac{3}{x}-\frac{3}{x^3})}}\\
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&&\\
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& = & \displaystyle{\frac{\lim_{x\rightarrow \infty} -2 +\lim_{x\rightarrow \infty} \frac{2}{x^2} +\lim_{x\rightarrow \infty} \frac{3}{x^3}}{\lim_{x\rightarrow \infty} 3+\lim_{x\rightarrow \infty} \frac{3}{x}-\lim_{x\rightarrow \infty}\frac{3}{x^3}}} \\
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&&\\
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& = & \displaystyle{\frac{-2+0+0}{3+0+0}}\\
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&&\\
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& = & \displaystyle{\frac{-2}{3}.}
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\end{array}</math>
 
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|&nbsp; &nbsp; '''(b)'''  &nbsp; &nbsp; <math>\frac{2}{3}</math>
 
|&nbsp; &nbsp; '''(b)'''  &nbsp; &nbsp; <math>\frac{2}{3}</math>
 
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|'''(c)'''  
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|&nbsp; &nbsp; '''(c)''' &nbsp; &nbsp; <math>\frac{-2}{3}</math>
 
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[[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 14:01, 17 February 2017

Find the following limits:

a) If 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 3} f(x).}
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{\tan(4x)}{\sin(6x)}. }
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 \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. }


Foundations:  
1. Linearity rules of limits
2. lim sin(x)/x


Solution:

(a)

Step 1:  
First, 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 3} \bigg(\frac{f(x)}{2x}+1\bigg)}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+\lim_{x\rightarrow 3} 1}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+1.} \end{array}}
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 \lim_{x\rightarrow 3} \frac{f(x)}{2x}=1.}
Step 2:  
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} 2x=6\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{1} & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 3} f(x)}{\lim_{x\rightarrow} 2x}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 3} f(x)}{6}.} \end{array}}

Multiplying both sides 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 6,} 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 3} f(x)=6.}

(b)

Step 1:  
First, we write
        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 0} \frac{\tan(4x)}{\sin(6x)}} & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(4x)}{\cos(4x)} \frac{1}{\sin(6x)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 0} \frac{4}{6} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}}\\ &&\\ & = & \displaystyle{\frac{4}{6}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}.} \end{array}}
Step 2:  
Now, 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 0} \frac{\tan(4x)}{\sin(6x)}} & = & \displaystyle{\frac{4}{6}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}}\\ &&\\ & = & \displaystyle{\frac{4}{6}\bigg(\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{6x}{\sin(6x)}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{1}{\cos(4x)}\bigg)}\\ &&\\ & = & \displaystyle{\frac{4}{6} (1)(1)(1)}\\ &&\\ & = & \displaystyle{\frac{2}{3}.} \end{array}}

(c)

Step 1:  
First, 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 \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}} & = & \displaystyle{\lim _{x\rightarrow \infty} \frac{(-2x^3-2x+3)}{(3x^3+3x^2-3)} \frac{(\frac{1}{x^3})}{(\frac{1}{x^3})}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 0} \frac{-2-\frac{2}{x^2}+\frac{3}{x^3}}{3+\frac{3}{x}-\frac{3}{x^3}}}. \end{array}}
Step 2:  
Now, we use the properties of limits to 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 \begin{array}{rcl} \displaystyle{\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}} & = & \displaystyle{\lim_{x\rightarrow \infty} \frac{-2-\frac{2}{x^2}+\frac{3}{x^3}}{3+\frac{3}{x}-\frac{3}{x^3}}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow \infty} (-2-\frac{2}{x^2}+\frac{3}{x^3})}{\lim_{x\rightarrow \infty} (3+\frac{3}{x}-\frac{3}{x^3})}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow \infty} -2 +\lim_{x\rightarrow \infty} \frac{2}{x^2} +\lim_{x\rightarrow \infty} \frac{3}{x^3}}{\lim_{x\rightarrow \infty} 3+\lim_{x\rightarrow \infty} \frac{3}{x}-\lim_{x\rightarrow \infty}\frac{3}{x^3}}} \\ &&\\ & = & \displaystyle{\frac{-2+0+0}{3+0+0}}\\ &&\\ & = & \displaystyle{\frac{-2}{3}.} \end{array}}


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 \lim_{x\rightarrow 3} f(x)=6}
    (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{2}{3}}
    (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 \frac{-2}{3}}

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