# How do you find the vertical, horizontal or slant asymptotes for (5e^x)/((e^x)-6)?

Feb 11, 2017

The vertical asymptote is $x = \ln 6$
The horizontal asymptote is $y = 5$ when x in ]ln6, +oo[
The horizontal asymptote is $y = 0$ when x in ]-oo, ln6[
No slant asymptote

#### Explanation:

Let $f \left(x\right) = \frac{5 {e}^{x}}{{e}^{x} - 6}$

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{\ln 6\right\}$

As you cannot divide by $0$, ${e}^{x} \ne 6$

The vertical asymptote is $x = \ln 6$

As the degree of the numerator is $=$ to the degree of the denominator, there is no slant asymptote.

${\lim}_{x \to + \infty} f \left(x\right) = {\lim}_{x \to + \infty} \frac{5 {e}^{x}}{e} ^ x = 5$

${\lim}_{x \to - \infty} f \left(x\right) = {\lim}_{x \to - \infty} \frac{5}{1 - \frac{6}{e} ^ x} = {0}^{-}$

The horizontal asymptote is $y = 5$ when x in ]ln6, +oo[

The horizontal asymptote is $y = 0$ when x in ]-oo, ln6[

graph{(5e^x)/(e^x-6) [-36.53, 36.54, -18.27, 18.28]}