Asymptotic expansion of $(\sin x)^x$ $(x\to0^+)$

Question

Determine the asymptotic expansion of $(\sin x)^x$ as $x\to0^+$. It is a question in my textbook, and I cannot find out how to calculate it. How can I solve this? The answer of the textbook is $$1+x\log x+\frac12\left(x^2 (\log x)^2\right)+o\!\left(x^2 (\log x)^2\right).$$

Answer 1

Let $f(x)= (\sin x)^x$ and $g(x) = \log f(x)$

Then $$g(x) = x \log(\sin(x)) = x \log (x + o(x^2)) = x \log (x(1 + o(x))\\ =x\log x +x \log(1+ o(x))\\ = x\log x +o(x^2) $$

Hence $g(x)\to 0$ as $x\to 0$

And $$f(x) = e^{g(x)} = 1 + g(x) + \frac{1}{2}g(x)^2 + o(g(x)^2) = \\ 1 + x\log x + \frac12 (x \log x)^2 + o((x \log x)^2)+o(x^2)=\\ 1 + x\log x + \frac12 (x \log x)^2 + o((x \log x)^2) $$