How do you find all the asymptotes for function $f (x) = \frac{17 x}{2 x^{2} + 3}$ $f(x)= (17x)/( 2x^2 + 3)$ ?

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How do you find all the asymptotes for function $f (x) = \frac{17 x}{2 x^{2} + 3}$ $f(x)= (17x)/( 2x^2 + 3)$ ?

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Answer 1 · 2015-09-24T15:35:25

Its horizontal asymptote is the $x$ $x$ -axis ( $y = 0$ $y=0$ ) as $x \to \pm \infty$ $x-> pm infty$ and it has no vertical asymptotes.

Explanation:

The line $y = 0$ $y=0$ is a horizontal asymptote of the rational function $f (x) = \frac{17 x}{2 x^{2} + 3}$ $f(x)=(17x)/(2x^2+3)$ since the degree of the bottom (denominator) is greater than the degree of the top (numerator). Furthermore, the denominator $2 x^{2} + 3$ $2x^2+3$ has no real roots (the roots are the imaginary numbers $\pm i \sqrt{\frac{3}{2}}$ $pm i sqrt{3/2}$ ). Hence, the function has no vertical asymptotes.

To confirm the horizontal asymptote, you can do the following limit calculation, which I'm not rigorously justifying:

$lim_{x-> pm infty}(17x)/(2x^2+3)=lim_{x-> pm infty}(17x)/(2x^2+3) * (1/x^2)/(1/x^2)$

$=lim_{x-> pm infty}(17/x)/(2+3/x^2)=(lim_{x->pm infty}(17/x))/(lim_{x->pm infty} 2+lim_{x->pm infty}(3/x^2))=0/(2+0)=0$

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How do you find all the asymptotes for function $f (x) = \frac{17 x}{2 x^{2} + 3}$ $f(x)= (17x)/( 2x^2 + 3)$ ?

1 Answer

Explanation:

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How do you find all the asymptotes for function f(x)= (17x)/( 2x^2 + 3)f(x)=17x2x2+3f(x)= (17x)/( 2x^2 + 3)?

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How do you find all the asymptotes for function $f (x) = \frac{17 x}{2 x^{2} + 3}$ $f(x)= (17x)/( 2x^2 + 3)$ ?