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

Oct 23, 2016

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

#### Explanation:

We start by factorising the denominator
${x}^{2} + x - 72 = \left(x - 8\right) \left(x + 9\right)$

As we cannot divide by so $x \ne 8$ and $x \ne - 9$
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 $x \to \pm \infty$
We take the highest order of the polynomials

$\lim y = \frac{8}{1}$
$x \to - \infty$

And
$\lim y = \frac{8}{1}$
$x \to \infty$
So the horizontal asymptote is $y = 8$