Calculate the limit $$\lim_{n\to\infty} \int_0^n\left(1+\frac xn\right)^ne^{-2x}\,dx$$
In class we've seen two results, the dominated and monotone convergence theorems.
For both we need that $\lim_{n\to\infty}f_n$ exists almost everywhere. In this case, we have $\lim_{n\to\infty}f_n(x)=e^xe^{-2x}=e^{-x}$.
But I haven't been able to find any integrable function $g$, such that $|f_n(x)|\le g(x)$ and use the dominated convergence theorem, or a way to see if $f_n$ is non-decreasing and use the monotone convergence theorem.
I'm now also thinking if it's possible or useful to do that, since there's an $n$ also in the integration domain.