Difference between revisions of "009A Sample Final A, Problem 6"

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! Foundations:    
 
! Foundations:    
 
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|Vertical asymptotes occur whenever the denominator of a rational function goes to zero, <u>''and''</u> it doesn't cancel from the numerator.
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|Vertical asymptotes occur whenever the denominator of a rational function goes to zero, <u>''and''</u> &thinsp;it doesn't cancel from the numerator.
 
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|On the other hand, horizontal asymptotes represent the limit as ''x'' goes to either positive or negative infinity.
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|On the other hand, horizontal asymptotes represent the limit as <math style="vertical-align: 0%;">x</math> goes to either positive or negative infinity.
 
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|&nbsp;&nbsp;&nbsp;&nbsp; <math>0 = 10x-20 = 10(x-2),</math>
 
|&nbsp;&nbsp;&nbsp;&nbsp; <math>0 = 10x-20 = 10(x-2),</math>
 
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|which has a root at ''x'' = 2.  This is our vertical asymptote.
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|which has a root at <math style="vertical-align: 0%;">x = 2.</math> This is our vertical asymptote.
 
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!Horizontal Asymptotes: &nbsp;
 
!Horizontal Asymptotes: &nbsp;
 
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|More work is required here.  Since we need to find the limits at <math style="vertical-align: -5%;">\pm\infty</math>, we can multiply our ''f''(''x'') by
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|More work is required here.  Since we need to find the limits at <math style="vertical-align: 0%;">\pm\infty</math>, we can multiply our <math style="vertical-align: -20%;">f(x)</math> by
 
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|&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{\sqrt{\frac{1}{x^{2}}}}{\,\,\,\frac{1}{x}}.</math>
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|<br>&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{\sqrt{\frac{1}{x^{2}}}}{\,\,\,\frac{1}{x}}.</math>
 
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|This expression is equal to 1 for positive values of ''x'', and is equal to -1 for negative values of ''x''.  Since multiplying ''f''(''x'') by an expression equal to 1 doesn't change the limit, we will add a negative sign to it when considering the limit as x goes to <math style="vertical-align: -5%;">-\infty</math>.  Thus,
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|<br>This expression is equal to <math style="vertical-align: -2%;">1</math> for positive values of <math style="vertical-align: 0%;">x</math>, and is equal to <math style="vertical-align: -3%;">-1</math> for negative values of <math style="vertical-align: 0%;">x</math>.  Since multiplying <math style="vertical-align: -20%;">f(x)</math> by an expression equal to <math style="vertical-align: -2%;">1</math> doesn't change the limit, we will add a negative sign to our fraction when considering the limit as <math style="vertical-align: 0%;">x</math> goes to <math style="vertical-align: -2%;">-\infty</math>.  Thus,
 
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|<br>&nbsp;&nbsp;&nbsp;&nbsp;  <math>\lim_{x\rightarrow\pm\infty}\frac{\sqrt{4x^{2}+3}}{10x-20}\,\,\cdot\,\,\pm\frac{\sqrt{\frac{1}{x^{2}}}}{\,\,\,\frac{1}{x}}=\lim_{x\rightarrow\pm\infty}\pm\frac{\sqrt{\frac{4x^{2}}{x^{2}}+\frac{3}{x^{2}}}}{\frac{10x}{x}-\frac{20}{x}} = \lim_{x\rightarrow\pm\infty}\pm\frac{\sqrt{4+\frac{3}{x^{2}}}}{10-\frac{20}{x}}=\pm\frac{2}{10}=\pm\frac{1}{5}</math>
 
|<br>&nbsp;&nbsp;&nbsp;&nbsp;  <math>\lim_{x\rightarrow\pm\infty}\frac{\sqrt{4x^{2}+3}}{10x-20}\,\,\cdot\,\,\pm\frac{\sqrt{\frac{1}{x^{2}}}}{\,\,\,\frac{1}{x}}=\lim_{x\rightarrow\pm\infty}\pm\frac{\sqrt{\frac{4x^{2}}{x^{2}}+\frac{3}{x^{2}}}}{\frac{10x}{x}-\frac{20}{x}} = \lim_{x\rightarrow\pm\infty}\pm\frac{\sqrt{4+\frac{3}{x^{2}}}}{10-\frac{20}{x}}=\pm\frac{2}{10}=\pm\frac{1}{5}</math>
 
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|<br>Thus, we have a horizontal asymptote at ''y'' = -1/5 on the left (as ''x'' goes to <math style="vertical-align: -5%;">-\infty</math>), and a horizontal asymptote at ''y'' = 1/5 as ''x'' goes to  <math style="vertical-align: -5%;">+\infty</math>).  
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|<br>Thus, we have a horizontal asymptote at <math style="vertical-align: -21%;">y=-1/5</math> on the left (as <math style="vertical-align: 0%;">x</math> goes to <math style="vertical-align: -2%;">-\infty</math>), and a horizontal asymptote at <math style="vertical-align: -22%;">y=1/5</math> on the right (as <math style="vertical-align: 0%;">x</math> goes to  <math style="vertical-align: -4%;">+\infty</math>).  
 
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[[009A_Sample_Final_A|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_A|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 19:35, 27 March 2015


6. Find the vertical and horizontal asymptotes of the function  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 f(x)=\frac{\sqrt{4x^{2}+3}}{10x-20}.}

Foundations:  
Vertical asymptotes occur whenever the denominator of a rational function goes to zero, and  it doesn't cancel from the numerator.
On the other hand, horizontal asymptotes represent the limit as 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} goes to either positive or negative infinity.

 Solution:

Vertical Asymptotes:  
Setting the denominator to zero, 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 0 = 10x-20 = 10(x-2),}
which has a root at 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.} This is our vertical asymptote.
Horizontal Asymptotes:  
More work is required here. Since we need to find the limits at 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 \pm\infty} , we can multiply our 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 f(x)} 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 \frac{\sqrt{\frac{1}{x^{2}}}}{\,\,\,\frac{1}{x}}.}

This expression is equal to 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} for positive values of 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} , and is equal to 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} for negative values of 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} . Since multiplying 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 f(x)} by an expression equal to 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} doesn't change the limit, we will add a negative sign to our fraction when considering the limit as 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} goes to 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 -\infty} . Thus,

     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\pm\infty}\frac{\sqrt{4x^{2}+3}}{10x-20}\,\,\cdot\,\,\pm\frac{\sqrt{\frac{1}{x^{2}}}}{\,\,\,\frac{1}{x}}=\lim_{x\rightarrow\pm\infty}\pm\frac{\sqrt{\frac{4x^{2}}{x^{2}}+\frac{3}{x^{2}}}}{\frac{10x}{x}-\frac{20}{x}} = \lim_{x\rightarrow\pm\infty}\pm\frac{\sqrt{4+\frac{3}{x^{2}}}}{10-\frac{20}{x}}=\pm\frac{2}{10}=\pm\frac{1}{5}}

Thus, we have a horizontal asymptote at 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 y=-1/5} on the left (as 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} goes to 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 -\infty} ), and a horizontal asymptote at 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 y=1/5} on the right (as 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} goes to 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 +\infty} ).

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