How do you find vertical, horizontal and oblique asymptotes for #(3x+5)/ (x-6)#?

1 Answer
Oct 28, 2016

Answer:

The vertical asymptote is #x=6#
The horizontal asymptote is #y=3#
There are no oblique asymptote

Explanation:

The function is not defined when #x=6# as we cannot divide by zero
So we have a vertical asymptote at #x=6#

As the degree of the numerator is identical to the degree of the denominator, so we make a long division

#3x+5##color(white)(aaaa)##∣##x-6#
#3x-18##color(white)(aaa)##∣##3#
#0-23#

Finally we obtain
#(3x+5)/(x-6)=3+23/(x-6)#
So #y=3# is a horizontal asymptote

We could get the same result by finding the limit as #x->+-oo#
limit #(3x+5)/(x-6)=3#
#x->+-oo#