I am realy stuck in solving the following limit problem. Can you find any function $g(x)$ by which $$\lim_{a\rightarrow \infty} \frac{a^N}{\log a} \int_{0}^\infty \frac{e^{-x}}{(1+ag(x))^N}dx = c$$ where $c$ is a nonzero constant.

The solution to this problem may contain some general properties on $g(x)$. But I can't even find a specific $g(x)$ for a specific $N$, say $N=2$.