Evaluating the limit $\lim\limits_{x\to 0+}\left[h(x)\right]^{g(x)}$

Question

Evaluate the limit $$\lim_{x\to 0+}\left[h(x)\right]^{g(x)}$$

where $h(x)=\log_{\ |\sin(x)|} x$ , $g(x)=f(x)+|x|$, and

$$f(x) = \begin{cases} x & x \leq 0 \\ -x & x >0 \end{cases}$$

Answer 1

It is clear that $\;f(x)=-|x|\;$ and thus $\;g(x)\equiv0\;$ , so $$\;h(x)^{g(x)}=h^0=1\;,\;\;\forall\,x\;\;\text{such that}\;\;x\neq0$$

and the limit then equals $\;1\;$ .