# How do you find the Vertical, Horizontal, and Oblique Asymptote given (6e^x)/(e^x-8)?

Dec 1, 2016

The vertical asymptote is $x = \ln 8$
The horizontal asymptotes are $y = 0$ and $y = 6$
No oblique asymptote

#### Explanation:

As you cannot divide by $0$

The denominator must be $\ne 0$

${e}^{x} - 8 \ne 0$

${e}^{x} \ne 8$

$x \ne \ln 8$

So the vertical asymptote is $x = \ln 8$

Let $f \left(x\right) = \frac{6 {e}^{x}}{{e}^{x} - 8}$

$f \left(x\right) = \frac{6 {e}^{x}}{{e}^{x} - 8} = \frac{6}{1 - 8 {e}^{- x}}$

${\lim}_{x \to - \infty} f \left(x\right) = {\lim}_{x \to - \infty} \frac{6}{\left(1 - 8 {e}^{- x}\right)} = \frac{6}{-} \infty = {0}^{-}$

${\lim}_{x \to + \infty} f \left(x\right) = {\lim}_{x \to + \infty} \frac{6}{\left(1 - 8 {e}^{- x}\right)} = 6$

The horizontal asymptotes are $y = 0$ and $y = 6$

graph{(6e^x)/(e^x-8) [-15.04, 16.99, -5.05, 10.96]}