# How to find the asymptotes for (3x^2) / (x^2-4)?

Aug 11, 2018

$\frac{3 {x}^{2}}{{x}^{2} - 4}$ has horizontal asymptote $y = 3$ and vertical asymptotes $x = - 2$ and $x = 2$

#### Explanation:

Note that:

${\lim}_{x \to \pm \infty} \frac{3 {x}^{2}}{{x}^{2} - 4} = {\lim}_{x \to \pm \infty} \frac{3}{1 - \frac{4}{x} ^ 2} = 3$

So $y = 3$ is a horizontal asymptote.

Also:

${x}^{2} - 4 = \left(x - 2\right) \left(x + 2\right)$

So the denominator is zero when $x = \pm 2$ and the numerator is non-zero at those values.

So $\frac{3 {x}^{2}}{{x}^{2} - 4}$ has vertical asymptotes at $x = - 2$ and $x = 2$

graph{(y-(3x^2)/(x^2-4)) = 0 [-10.09, 9.91, -3.6, 6.4]}