Question

What are the asymptotes for `(x + 4)/(x^2+x-6)`?

Answer 1

In cases like this, you must often rewrite (factorise)

Explanation:

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

Since the numerator may not be `=0` we conclude:
`x!=2andx!=-3`

So `x=2andx=-3` are the vertical asymptotes.

As `x` gets larger, the `4,-2and3` makes less and less of a difference, so it begins to look like:
`x/(x*x)=1/x` which will get smaller and smaller.

So `y=0` is the horizontal asymptote.
graph{(x+4)/(x^2+x-6) [-10, 10, -5, 5]}