Question

How do you find all the asymptotes for function `F(x)=(x^2+x-12)/(x^2-4)`?

Answer 1

We first factorize:

Explanation:

`x^2+x-12` may be written as `(x-4)(x+3)` and
`x^2-4` may be written as `(x-2)(x+2)`

If either of the denominator-factors get closer to `0` we have a vertical asymptote or a "hole".

The whole function now becomes:

`F(x)=((x-4)(x+3))/((x-2)(x+2))`

Which cannot be further reduced.
So the vertical asymtotes are at `x=-2 and x=+2`

Since the grade and factor of the highest power are the same, the function will tend to `1` for very high values of `x`
So the horizontal asymptote is `y=1`

graph{(x^2+x-12)/(x^2-4) [-28.9, 28.85, -14.43, 14.45]}