Question

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

Answer 1

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.