This is a question from Calculus 7e by Stewart:
It's easy to prove by calculating the limit of $e^{ln(f(x))\cdot g(x)}$ if $f(x)$ is restricted to a positive function. However, seems that if we remove this restriction the conclusion still holds. But then we cannot use $ln(f(x))$ since $f(x)\leq 0$ is possible. So how to prove the general form of this limit?
In calculus context for function of real variables, the function
$$[f(x)]^{g(x)}$$
is restricted and defined only for $f(x)>0$.
Refer also to the related
Let $\varepsilon>0$. Then $\min\left\{\varepsilon,\frac12\right\}>0$ and therefore there is a $\delta_1>0$ such that $$0<\lvert x-a\rvert<\delta_1\implies\bigl\lvert f(x)\bigr\rvert<\min\left\{\varepsilon,\frac12\right\}.$$And there is a $\delta_2>0$ such that$$0<\lvert x-a\rvert<\delta_2\implies g(x)>1.$$Therefore, if $\delta=\min\{\delta_1,\delta_2\}$,$$0<\lvert x-a\rvert<\delta\implies\left\lvert f(x)^{g(x)}\right\rvert=\bigl\lvert f(x)\bigr\rvert^{g(x)}<\min\left\{\varepsilon,\frac12\right\}\leqslant\varepsilon.$$
