Question

How do you find vertical, horizontal and oblique asymptotes for `( x²-x-12 )/(x+5)`?

Answer 1

The vertical asymptote is `x=-5`
The oblique asymptote is `y=x-6`

Explanation:

The domain of the function is `RR-(5)`
as we cannot divide by `0`
So the vertical asymptote is `x=-5`
The degree of the polynomial of the numerator `>` degree of the denominator, so we have an oblique asymptote
doing a long division give

`(x^2-x-12)/(x+5)=x-6+(18)/(x+5)`

So `y=x-6` is an oblique asymptote

Verification

`x-6+(18)/(x+5)=((x-6)(x+5)+18)/(x+5)=(x^2-x-30+18)/(x+5)`

`=(x^2-x-12)/(x+5)`