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

1 Answer
Sep 25, 2014

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:

Graph of a slant asymptote.

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.