Limit of x to the x/Proof 2
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Theorem
Let $f: \R_{>0} \to \R$ be defined as:
- $\forall x \in \R_{>0}: \map f x = x^x$
Then:
- $\ds \lim_{x \mathop \to 0^+} x^x = 1$
Proof
\(\ds \lim_{x \mathop \to 0^+} x^x\) | \(=\) | \(\ds \lim_{x \mathop \to 0^+} \map \exp {x \ln x}\) | Definition 1 of Power (Algebra) | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \exp {\lim_{x \mathop \to 0^+} x \ln x}\) | Exponential Function is Continuous | |||||||||||
\(\ds \) | \(=\) | \(\ds \exp 0\) | Limit of Power of $x$ by Absolute Value of Power of Logarithm of $x$: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | Exponential of Zero |
$\blacksquare$