# How do you find the vertical, horizontal or slant asymptotes for y= x / e^x?

Nov 16, 2016

The horizontal asymptote is $y = 0$
No slant or vertical asymptote.

#### Explanation:

The exponential function ${e}^{x}$ is always positive.

$\forall x \in \mathbb{R} , {e}^{x} > 0$

So we don't have a vertical asymptote:

${\lim}_{x \to - \infty} y = \frac{x}{e} ^ \left(- \infty\right) = \left(- \infty \cdot {e}^{\infty}\right) = - \infty$

${\lim}_{x \to + \infty} y = \frac{x}{e} ^ \left(+ \infty\right) = \left(\frac{x}{e} ^ \left(\infty\right)\right) = {0}^{+}$

We have a horizontal asymptote $y = 0$

graph{(y-x/e^x)(y)=0 [-6.29, 7.757, -4.887, 2.136]}