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

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

Question
Limits and continuity
asked 2021-02-25
Evaluate the following limits. \(\lim_{x\rightarrow0^+}x^{x^2}\)

Answers (1)

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
0

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