Limit of $x^{x}$ as $x \to 0^{+}$

The limit of x^x as x → 0^+ is 1.

The limit of $x^{x}$ as $x \to 0^{+}$ is $1$ .

\lim_{x \to 0^{+}} x^{x} = 1

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

How to work it out

1
We want what $x^{x}$ approaches as x → 0⁺.
$\lim_{x \to 0^{+}} x^{x}$
2
Direct substitution gives an indeterminate form: $0^{0}$ . It does not tell us the limit.
3
Method: $x^{x}$ = $e^{x ln x}$ and x ln x → 0.
4
Result:
$\lim_{x \to 0^{+}} x^{x} = 1$

LaTeX source of the formulas on this page

\[ \lim_{x \to 0^{+}} x^{x} = 1 \]
\[ \lim_{x \to 0^{+}} x^{x} \]

What is the limit of

x^{x}

as x approaches 0⁺?

The limit of $x^{x}$ as $x \to 0^{+}$ is $1$ .

Which form is it?

$0^{0}$ .