How do you find the slant asymptote of y=(x^3)/((x^2)-3)?

Aug 14, 2018

The slant asymptote is: $y = x$

Explanation:

Given:

$y = {x}^{3} / \left({x}^{2} - 3\right)$

Note that:

${x}^{3} / \left({x}^{2} - 3\right) = \frac{{x}^{3} - 3 x + 3 x}{{x}^{2} - 3}$

$\textcolor{w h i t e}{{x}^{3} / \left({x}^{2} - 3\right)} = \frac{x \left({x}^{2} - 3\right) + 3 x}{{x}^{2} - 3}$

$\textcolor{w h i t e}{{x}^{3} / \left({x}^{2} - 3\right)} = x + \frac{3 x}{{x}^{2} - 3}$

and:

${\lim}_{x \to \pm \infty} \frac{3 x}{{x}^{2} - 3} = {\lim}_{x \to \pm \infty} \frac{3}{x - \frac{3}{x}} = {\lim}_{x \to \pm \infty} \frac{3}{x} = 0$

So:

$y = {x}^{3} / \left({x}^{2} - 3\right)$

is asymptotic to $y = x$