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

Question
Limits and continuity
Evaluate the following limits. $$\lim_{x\rightarrow0^+}x^{x^2}$$

2021-02-26
We have given:
$$\lim_{x\rightarrow0^+}x^{x^2}$$
By the exponent rule a^x=e^{(\ln(a^x)}=e^{x\ln a}\)
$$\lim_{x\rightarrow0^+}x^{x^2}=\lim_{x\rightarrow0^+}ex^{2\ln(x)}$$
By the chain rule,
$$\lim_{x->0^+}x^{x^2}=\lim_{x\rightarrow0^+}e^{x^2\cdot x}$$
$$\lim_{x\rightarrow0^+}x^{x^2}=\lim_{x\rightarrow0^+}e^{x^2}+2x^2e^{x^2}$$
$$\lim_{x\rightarrow0^+}x^{x^2}=e^0+0$$
$$\lim_{x\rightarrow0^+}x^{x^2}=1+0$$
$$\lim_{x\rightarrow0^+}x^{x^2}=1$$
Result: 1

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