Limit of x to the x

Theorem
Let $f: \R_{>0} \to \R$ be defined as:
 * $\forall x \in \R_{>0}: \map f x = x^x$

Then:
 * $\ds \lim_{x \mathop \to 0^+} x^x = 1$

Also presented as
This can also be presented as:
 * $\ds \lim_{x \mathop \to 0^+} \map \exp {x \ln x} = 1$

as is demonstrated during the course of the proof.