Let $f$, $g$ be two continuous function.
I want to calculate $$ \phi(x)=\lim_{\beta\rightarrow \infty} \frac{\int_{x\in[-1,1]}[e^{\beta f(x)}g(x)] dx}{\int_{x\in[-1,1]}[e^{\beta f(x)}] dx}. $$
If $f\equiv g$, I know the value of the limit is $\max_{x\in[-1,1]}f(x).$
What if $f\neq g$? Is the answer $$\max\{g(x): x\in\arg\max f(x)\}$$ right?
For example, Let $f(x)= xy$, and $g(x)\equiv y$ for some $y\in \mathbb{R}$. In this case, the value is 1, which equals $\max\{g(x): x\in\arg\max f(x)\}$.
What if $x$ becomes a vector, $f$ becomes $f: \mathbb{R}^n\mapsto \mathbb{R}$ and $g$ becomes $g: \mathbb{R}^n\mapsto \mathbb{R}^n$? Would the answer be $$ (\phi(x))_i=\max\{g_i(x): x\in\arg\max f(x)\} \text{ for } i=1,2,\ldots,n? $$
Furthermore, which one of the following is bigger with a large $\beta$, $$ \frac{\int_{x\in[-1,1]}[e^{\beta f(x)}f(x)] dx}{\int_{x\in[-1,1]}[e^{\beta f(x)}] dx}, \text{ and } \log\left(\int_{x\in[-1,1]}[e^{\beta f(x)}] dx\right)/\beta. $$