Question

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

Answer 1

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(x) = (2x+4)/(x^2-3x-4) = (2(x+2))/((x-4)(x+1))`

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(x) -> 0` as `x -> +-oo`

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

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