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

\(\lim_{x\rightarrow0^+}x^{x^2}\)

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