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

Question

Evaluate the following limits.

lim_{x \to 0^{+}} x^{x^{2}}

Answer 1

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

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

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