Difference between revisions of "009A Sample Final 2, Problem 8"

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<span class="exam">(c) &nbsp;<math style="vertical-align: -15px">\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}</math>
 
<span class="exam">(c) &nbsp;<math style="vertical-align: -15px">\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}</math>
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[[009A Sample Final 2, Problem 8 Solution|'''<u>Solution</u>''']]
  
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!Foundations: &nbsp;
 
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|<span class="exam">In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.
 
  
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<span class="exam">(a) &nbsp; <math style="vertical-align: -14px">\lim_{x\rightarrow -3} \frac{x^3-9x}{6+2x}</math>
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[[009A Sample Final 2, Problem 8 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
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<span class="exam">(b) &nbsp; <math style="vertical-align: -14px">\lim_{x\rightarrow 0^+} \frac{\sin (2x)}{x^2}</math>
 
  
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<span class="exam">(c) &nbsp; <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{3x}{\sqrt{4x^2+x+5}}</math>
 
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'''Solution:'''
 
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'''(a)'''
 
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!Step 1: &nbsp;
 
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!Step 2: &nbsp;
 
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'''(b)'''
 
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!Step 1: &nbsp;
 
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'''(c)'''
 
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!Step 1: &nbsp;
 
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|We proceed using L'Hôpital's Rule. So, we have
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 
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\displaystyle{\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}} & \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow 1}\frac{3x^2}{10x^9}.}
 
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\end{array}</math>
 
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!Step 2: &nbsp;
 
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|Now, we have
 
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 
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\displaystyle{\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}} & \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow 1}\frac{3x^2}{10x^9}}\\
 
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&&\\
 
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& = & \displaystyle{\frac{3(1)^2}{10(1)^9}}\\
 
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&&\\
 
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& = & \displaystyle{\frac{3}{10}.}
 
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\end{array}</math>
 
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!Final Answer: &nbsp;
 
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|'''(a)'''
 
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|'''(b)'''
 
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|&nbsp; &nbsp;'''(c)'''&nbsp; &nbsp; <math>\frac{3}{10}</math>
 
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[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 09:23, 1 December 2017

Compute

(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 \infty} \frac{x^{-1}+x}{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 0} \frac{\sin x}{\cos x-1}}

(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 1} \frac{x^3-1}{x^{10}-1}}

Solution


Detailed Solution


Return to Sample Exam

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