How do you find the Vertical, Horizontal, and Oblique Asymptote given #R(x) = (3x) / (x^2 - 9)#?

1 Answer
Nov 17, 2016

Answer:

Vertical asymptotes: #x=-3#, #x=3#
Horizontal asymptote: #y=0#
Oblique asymptote: none.

Explanation:

1) To find the vertical asymptotes factor the denominator and see what values of #x# make a factored expression equal to zero, these values are the vertical asymptotes.

#x^2-9=(x+3)(x-3)#
The first vertical asymptote: #x+3=0-> #"#x=-3#"
The second vertical asymptote: #x-3=0->#"#x=3#"

2) To find the horizontal asymptote divide the highest degree term in the numerator by the highest degree term in the denominator and see what would happen to #R(x)# as #x# goes to high values.

#(3x)/x^2=3/x#, if #x# goes to high values the number will reach zero, so the horizontal asymptote is zero.

3) There is no oblique asymptotes since #3x# is not divisible by #x^2-9#. There will be an oblique asymptote if the numerator is one degree higher than the denominator and the asymptote will be the quotient of the algebraic long division.