Limit of $\left(1 + x\right)^{\frac{1}{x}}$ as $x \to 0$

The limit of ≤ft(1 + x)^(1)/(x) as x → 0 is e.

The limit of $\left(1 + x\right)^{\frac{1}{x}}$ as $x \to 0$ is $e$ .

\lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}} = e

Graph of (1 + x)^(1 / x) near x = 0. The open circle marks the point x = 0, where the function may be undefined; the curve still heads to e.

How to work it out

1
We want what (1 + x)^(1 / x) approaches as x → 0.
$\lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}}$
2
Direct substitution gives an indeterminate form: 1^∞. It does not tell us the limit.
3
Method: the same limit as (1 + 1/n)ⁿ with x = 1/n.
4
Result:
$\lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}} = e$

LaTeX source of the formulas on this page

\[ \lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}} = e \]
\[ \lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}} \]

What is the limit of (1 + x)^(1 / x) as x approaches 0?

The limit of $\left(1 + x\right)^{\frac{1}{x}}$ as $x \to 0$ is $e$ .

Which form is it?

1^∞.