TITLE: Evaluating $\lim_{n \rightarrow \infty} \int^{n}_{0} (1+\frac{x}{n})^{-n} \log(2+ \cos(\frac{x}{n})) \> dx$
QUESTION [5 upvotes]: The problem I am stuck on asks the reader to find the following limit:
$$\lim_{n \rightarrow \infty} \int^{n}_{0} \left(1+\frac{x}{n}\right)^{-n} \log\left(2+ \cos\left(\frac{x}{n}\right)\right)\ \mathrm dx.$$
The section I am working on contains all your basic limit theorem in measure theory (Monotone Covergence Theorem, Fatou's Lemma, Dominated Convergence Theorem). I know I am probably overseeing an application of one of them. Help would be greatly appreciated.

REPLY [3 votes]: For $0 \leq y \leq 1$ we have $\log(1+y) \geq y - y^2/2$, so for $0 < x \leq n$ we have
$$
\begin{align}
\left(1+\frac{x}{n}\right)^{-n} &= \exp\left[ -n \log\left(1+\frac{x}{n}\right)\right] \\
&\leq \exp\left[-n \left(\frac{x}{n} - \frac{x^2}{2n^2}\right)\right] \\
&= \exp\left[-x + \frac{x^2}{2n}\right] \\
&\leq \exp\left[-x + \frac{x^2}{2x}\right] \\
&= e^{-x/2}.
\end{align}
$$
Now apply dominated convergence.