Is it possible to find the order of magnitude of a function $$f(n) = \log \int_{\mathbb{R}} \frac{1}{(n+e^x)^{n}}g(x)\, dx$$ as $n\in\mathbb{N}$ goes to $+\infty$?
Here $g(x)$ is some function of $x$, not involving $n$, that can not be integrated analytically. What I do not understand is if it possible to understand the limiting behavior even if the integral can not be computed.
If $(n + e^x)^{-n} g(x)$ is absolutely integrable for some positive value of $n$, then $$\lim_{n \to \infty} n^n \int_{\mathbb R} (n + e^x)^{-n} g(x) dx = \lim_{n \to \infty} \int_{\mathbb R} \left( 1 + \frac {e^x} n \right)^{-n} g(x) dx = \int_{\mathbb R} e^{-e^x} g(x) dx.$$