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

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(Created page with "<br><span style="font-size:135%"><font face=Times Roman>6. Find the vertical and horizontal asymptotes of the function</font face=Times Roman> </span>  <math style="verti...")
 
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! Foundations: &nbsp;  
 
! Foundations: &nbsp;  
 
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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;'''Solution:'''
  
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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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  

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 goes to either positive or negative infinity.

 Solution:

Vertical Asymptotes:  
Setting the denominator to zero, we have
    
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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