Limit of some specific "almost Riemann" sums
For every $1\leqslant k\leqslant n$, $n^2\leqslant n^2+k\leqslant n^2+n$ hence the $n$th sum $S_n$ is such that $$ \sum_{k=1}^n\frac{k}{n^2+n}\leqslant S_n\leqslant\sum_{k=1}^n\frac{k}{n^2}. $$ Now, if it happens that the lower bound and the upper bound both converge to the same limit $\ell$, then $\lim\limits_{n\to\infty}S_n=\ell$.
Hence, what is left to do is to show that $\ell$ exists and to compute its value (or rather, to check that your guess that $\ell=\frac12$ is indeed correct--which it is).