Question

How do you find the vertical, horizontal or slant asymptotes for `y = (8 x^2 + x - 2)/(x^2 + x - 72)`?

Answer 1

The vertical asymptotes are `x=8` and `x=-9`
and the horizontal asymptote is `y=8`

Explanation:

We start by factorising the denominator
`x^2+x-72=(x-8)(x+9)`

As we cannot divide by so `x!=8` and `x!=-9`
So the vertical asymptotes are `x=8` and `x=-9`

As the degree of the polynomial of the numerator and denominator are the same, there is no slant asymptote.

To find the horizontal asymptote, we find the limit as `x->+-oo`
We take the highest order of the polynomials

`lim y=8/1`
`x->-oo`

And
`lim y=8/1`
`x->oo`
So the horizontal asymptote is `y=8`