Question

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

Answer 1

Unfortunately,

`y={3x^6-7x+10}/{8x^5+9x+10}`

does not have any horizontal asymptote; however, it has a slant asymptote `y=3/8x` (in green).

Its graph looks like this:

Let us look at some details.

`lim_{x to pm infty}{3x^6-7x+10}/{8x^5+9x+10}`

by dividing by `x^5`,

`=lim_{x to infty}{3x-7/x^4+10/x^5}/{8+9/x^4+10/x^5}`

`={pm infty-0+0}/{8+0+0}=pm infty`

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