Limit of $\frac{x^{2}}{e^{x}}$ as $x \to \infty$

Q: What is the limit of x^2 / exp(x) as x approaches ∞?

The limit of x^2e^x as x → is 0.

The limit of $\frac{x^{2}}{e^{x}}$ as $x \to \infty$ is $0$ .

\lim_{x \to \infty} \frac{x^{2}}{e^{x}} = 0

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Graph of x^2 / exp(x) near x = ∞. The curve levels off at 0 as x grows.

How to work it out

1
We want what $x^{2}$ / exp(x) approaches as x → ∞.
$\lim_{x \to \infty} \frac{x^{2}}{e^{x}}$
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.
$\lim \frac{x^{2}}{e^{x}} = \lim \frac{2 x}{e^{x}}$
4
Method: apply L'Hôpital twice.
5
Result:
$\lim_{x \to \infty} \frac{x^{2}}{e^{x}} = 0$

LaTeX source of the formulas on this page

\[ \lim_{x \to \infty} \frac{x^{2}}{e^{x}} = 0 \]
\[ \lim_{x \to \infty} \frac{x^{2}}{e^{x}} \]
\[ \lim \frac{x^{2}}{e^{x}} = \lim \frac{2 x}{e^{x}} \]

Values of $x^{2}$ / exp(x) as x → ∞ (computed)

x	f(x)	Distance from the limit
10	0.0045399930	4.54e-3
100	3.7200760e-40	3.72e-40
1000	0.0000000	0.00e+0
10000	0.0000000	0.00e+0
100000	0.0000000	0.00e+0
1000000	0.0000000	0.00e+0

Questions

What is the limit of

x^{2}

/ exp(x) as x approaches ∞?

The limit of $\frac{x^{2}}{e^{x}}$ as $x \to \infty$ is $0$ .

Which form is it?

∞/∞.

Limit of x2ex\frac{x^{2}}{e^{x}}exx2​ as x→∞x \to \inftyx→∞

How to work it out

Values of x2x^{2}x2 / exp(x) as x → ∞ (computed)

Questions

More well-known limits

Limit of $\frac{x^{2}}{e^{x}}$ as $x \to \infty$

Values of $x^{2}$ / exp(x) as x → ∞ (computed)