How do you find the Vertical, Horizontal, and Oblique Asymptote for #R(x)=(3x+5) \ (x-6)#?

2 Answers
Sep 15, 2015

Answer:

This function has no asymptotes.

Explanation:

The given function is a polynomial and polynomials do not have asymptotes.

Sep 15, 2015

Answer:

Was the question intended to be for the function #R(x) = (3x+5)/(x-6)#?

Explanation:

The quotient is already reduced. (There are no common factors onf the numerator and the denominator).
So we can find vertical asymptotes by solving

denominator = #0#.

The equation of the vertical asymptote is #x=6#.

For values of #x# far from #0# (positive or negative), we have

#R(x) = (cancel(x)(1+5/x))/(cancel(x)(1-6/x))#

For #x# far from #0# (#x# with large absolute value), both #5/x# and #6/x# are close to #0#, so #R(x)# is close to #3/1 = 3#.

The line #y=3# is a horizontal asymptote on both sides.

The graph of a rational function cannot have both horizontal and obliques asymptotes. The graph of this function does have horizontal asymptotes, so it cannot have oblique asymptotes.