# How do you find the horizontal asymptote of the graph of y=3x^6-7x+10/8x^5+9x+10?

Sep 25, 2014

Unfortunately,

$y = \frac{3 {x}^{6} - 7 x + 10}{8 {x}^{5} + 9 x + 10}$

does not have any horizontal asymptote; however, it has a slant asymptote $y = \frac{3}{8} x$ (in green).

Its graph looks like this:

Let us look at some details.

${\lim}_{x \to \pm \infty} \frac{3 {x}^{6} - 7 x + 10}{8 {x}^{5} + 9 x + 10}$

by dividing by ${x}^{5}$,

$= {\lim}_{x \to \infty} \frac{3 x - \frac{7}{x} ^ 4 + \frac{10}{x} ^ 5}{8 + \frac{9}{x} ^ 4 + \frac{10}{x} ^ 5}$

$= \frac{\pm \infty - 0 + 0}{8 + 0 + 0} = \pm \infty$

Since the limits at infinity do not exist, there are no horizontal asymptotes.