# Evaluate the following limits. lim_{xrightarrow0^+}x^{x^2}

Cem Hayes 2021-02-25 Answered
Evaluate the following limits. $\underset{x\to {0}^{+}}{lim}{x}^{{x}^{2}}$
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## Expert Answer

Ezra Herbert
Answered 2021-02-26 Author has 99 answers

We have given:
$\underset{x\to {0}^{+}}{lim}{x}^{{x}^{2}}$
By the exponent rule ${a}^{x}={e}^{\mathrm{ln}\left({a}^{x}\right)}={e}^{x\mathrm{ln}a}$
$\underset{x\to {0}^{+}}{lim}{x}^{{x}^{2}}=\underset{x\to {0}^{+}}{lim}e{x}^{2\mathrm{ln}\left(x\right)}$
By the chain rule,
$\underset{x->{0}^{+}}{lim}{x}^{{x}^{2}}=\underset{x\to {0}^{+}}{lim}{e}^{{x}^{2}\cdot x}$
$\underset{x\to {0}^{+}}{lim}{x}^{{x}^{2}}=\underset{x\to {0}^{+}}{lim}{e}^{{x}^{2}}+2{x}^{2}{e}^{{x}^{2}}$
$\underset{x\to {0}^{+}}{lim}{x}^{{x}^{2}}={e}^{0}+0$
$\underset{x\to {0}^{+}}{lim}{x}^{{x}^{2}}=1+0$
$\underset{x\to {0}^{+}}{lim}{x}^{{x}^{2}}=1$
Result: 1

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