Answer: C
------------------------------------------
$$\begin{align*}\lim_{n\to\infty}\frac{1}{n}\left(\frac{n}{n+1}+\frac{n}{n+2}+\cdots+\frac{n}{2n}\right)=&\lim_{n\to\infty}\sum_{r=1}^n\frac{1}{n}\left(\frac{n}{n+r}\right) \\=&\lim_{n\to\infty}\sum_{r=1}^n\frac{1}{n}\left(\frac{1}{1+\frac{r}{n}}\right)\\ =& \int_0^1\frac{1}{1+x}dx\\ =& \log_e2\end{align*}$$
Here, $\frac{r}{n}=x$ and $\frac{1}{n}=dx$.