# How do you find all the asymptotes for function (2x+4)/(x^2-3x-4)?

Sep 17, 2015

Factorise the denominator and examine the degrees of the numerator and denominator to find vertical asymptotes $x = - 1$, $x = 4$ and horizontal asymptote $y = 0$.

#### Explanation:

$f \left(x\right) = \frac{2 x + 4}{{x}^{2} - 3 x - 4} = \frac{2 \left(x + 2\right)}{\left(x - 4\right) \left(x + 1\right)}$

This will have vertical asymptotes $x = - 1$ and $x = 4$. For both of these values of $x$ the denominator is $0$ and the numerator is non-zero.

Since the degree $2$ of the denominator is greater than the degree $1$ of the numerator, we find:

$f \left(x\right) \to 0$ as $x \to \pm \infty$

So there is a horizontal asymptote $y = 0$.

graph{(2x+4)/(x^2-3x-4) [-10, 10, -5, 5]}