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. | + | |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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|On the other hand, horizontal asymptotes represent the limit as ''x'' goes to either positive or negative infinity. | |On the other hand, horizontal asymptotes represent the limit as ''x'' goes to either positive or negative infinity. | ||
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| <math>0 = 10x-20 = 10(x-2),</math> | | <math>0 = 10x-20 = 10(x-2),</math> | ||
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| − | |which has a root at | + | |which has a root at <math style="vertical-align: 0%;">x = 2.</math> This is our vertical asymptote. |
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!Horizontal Asymptotes: | !Horizontal Asymptotes: | ||
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| − | |More work is required here. Since we need to find the limits at <math style="vertical-align: | + | |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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| <math>\frac{\sqrt{\frac{1}{x^{2}}}}{\,\,\,\frac{1}{x}}.</math> | | <math>\frac{\sqrt{\frac{1}{x^{2}}}}{\,\,\,\frac{1}{x}}.</math> | ||
Revision as of 21:39, 26 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 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 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 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 y = -1/5 on the left (as 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 y = 1/5 as 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} ). |