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

Nov 17, 2016

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.

${x}^{2} - 9 = \left(x + 3\right) \left(x - 3\right)$
The first vertical asymptote: $x + 3 = 0 \to$"$x = - 3$"
The second vertical asymptote: $x - 3 = 0 \to$"$x = 3$"

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 $R \left(x\right)$ as $x$ goes to high values.

$\frac{3 x}{x} ^ 2 = \frac{3}{x}$, if $x$ goes to high values the number will reach zero, so the horizontal asymptote is zero.

3) There is no oblique asymptotes since $3 x$ 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.