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

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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
−
\displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}} & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}\frac{\big(\frac{1}{x^3}\big)}{\big(\frac{1}{x^3}\big)}}\\
+
\displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}} & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6(1-\frac{1}{9x^5})}}{3x^3+4x}}\\
 
&&\\
 
&&\\
−
& = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9-\frac{1}{x^5}}}{3+\frac{4}{x^2}}.}
+
& = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{3|x^3|\sqrt{(1-\frac{1}{9x^5})}}{3x^3+4x}}\\
 +
&&\\
 +
& = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{3(-x^3)\sqrt{(1-\frac{1}{9x^5})}}{3x^3(1+\frac{4}{3x^2})}}
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
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|-
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
−
\displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}} & = & \displaystyle{\frac{\lim_{x\rightarrow -\infty} \sqrt{9-\frac{1}{x^5}}}{\lim_{x\rightarrow -\infty}3+\frac{4}{x^2}}}\\
+
\displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}} & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{3(-x^3)\sqrt{(1-\frac{1}{9x^5})}}{3x^3(1+\frac{4}{3x^2})}}\\
 +
&&\\
 +
& = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{-\sqrt{(1-\frac{1}{9x^5})}}{1+\frac{4}{3x^2}}}\\
 
&&\\
 
&&\\
−
& = & \displaystyle{\frac{\sqrt{9}}{3}}\\
+
& = & \displaystyle{\frac{-\sqrt{1}}{1}}\\
 
&&\\
 
&&\\
−
& = & \displaystyle{1.}
+
& = & \displaystyle{-1.}\\
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
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|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; <math>-\frac{3}{4}</math>
 
|&nbsp; &nbsp;'''(b)'''&nbsp; &nbsp; <math>-\frac{3}{4}</math>
 
|-
 
|-
−
|&nbsp; &nbsp;'''(c)'''&nbsp; &nbsp; <math>1</math>
+
|&nbsp; &nbsp;'''(c)'''&nbsp; &nbsp; <math>-1</math>
 
|}
 
|}
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 09:07, 4 December 2017

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

(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 0} \frac{\sin(5x)}{1-\sqrt{1-x}}}

(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:  
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=0}   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{\sin(5x)}{1-\sqrt{1-x}},}
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 denominator.
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 0} \frac{\sin(5x)}{1-\sqrt{1-x}}} & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(5x)}{1-\sqrt{1-x}} \bigg(\frac{1+\sqrt{1-x}}{1+\sqrt{1-x}}\bigg)}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(5x)(1+\sqrt{1-x})}{x}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(5x)}{x}(1+\sqrt{1-x})}\\ &&\\ & = & \displaystyle{\bigg(\lim_{x\rightarrow 0} \frac{\sin(5x)}{x}\bigg) \lim_{x\rightarrow 0}(1+\sqrt{1-x})}\\ &&\\ & = & \displaystyle{\bigg(5\lim_{x\rightarrow 0} \frac{\sin(5x)}{5x}\bigg) (2)}\\ &&\\ & = & \displaystyle{5(1)(2)}\\ &&\\ & = & \displaystyle{10.} \end{array}}

(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{\displaystyle{\lim_{x\rightarrow 8} xf(x)}}{\displaystyle{\lim_{x\rightarrow 8} 3}}}\\ &&\\ & = & \displaystyle{\frac{\displaystyle{\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{-6} & = & \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:  
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 -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}} & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{\sqrt{9x^6(1-\frac{1}{9x^5})}}{3x^3+4x}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{3|x^3|\sqrt{(1-\frac{1}{9x^5})}}{3x^3+4x}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{3(-x^3)\sqrt{(1-\frac{1}{9x^5})}}{3x^3(1+\frac{4}{3x^2})}} \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 -\infty} \frac{\sqrt{9x^6-x}}{3x^3+4x}} & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{3(-x^3)\sqrt{(1-\frac{1}{9x^5})}}{3x^3(1+\frac{4}{3x^2})}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow -\infty} \frac{-\sqrt{(1-\frac{1}{9x^5})}}{1+\frac{4}{3x^2}}}\\ &&\\ & = & \displaystyle{\frac{-\sqrt{1}}{1}}\\ &&\\ & = & \displaystyle{-1.}\\ \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 10}
   (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)    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 -1}

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