Limit of $\frac{x}{e^{x}}$ as $x \to \infty$

The limit of xe^x as x → is 0.

The limit of $\frac{x}{e^{x}}$ as $x \to \infty$ is $0$ .

\lim_{x \to \infty} \frac{x}{e^{x}} = 0

Graph of x / exp(x) near x = ∞. The curve levels off at 0 as x grows.

How to work it out

1
We want what x / exp(x) approaches as x → ∞.
$\lim_{x \to \infty} \frac{x}{e^{x}}$
2
Direct substitution gives an indeterminate form: ∞/∞. It does not tell us the limit.
3
L'Hôpital's rule: when the form is ∞/∞, the limit equals the limit of the ratio of the derivatives.
$\lim \frac{x}{e^{x}} = \lim \frac{1}{e^{x}}$
4
Method: exponentials beat powers: L'Hôpital gives 1/eˣ → 0.
5
Result:
$\lim_{x \to \infty} \frac{x}{e^{x}} = 0$

LaTeX source of the formulas on this page

\[ \lim_{x \to \infty} \frac{x}{e^{x}} = 0 \]
\[ \lim_{x \to \infty} \frac{x}{e^{x}} \]
\[ \lim \frac{x}{e^{x}} = \lim \frac{1}{e^{x}} \]

What is the limit of x / exp(x) as x approaches ∞?

The limit of $\frac{x}{e^{x}}$ as $x \to \infty$ is $0$ .

Which form is it?

∞/∞.