If so, the answer would be $\frac{\pi}{2}$ which is NOT there in the stated options.
BTW $\displaystyle \lim_{n\to \infty} \sum_{k=0}^{n}\frac{n}{n^2+k^2}=\frac{\pi}{4}$
but $\displaystyle \sum_{k=0}^{\infty}\frac{n}{n^2+k^2}=\frac{\pi}{2}$
So definitely the question has to be
$\displaystyle \lim_{n \to \infty} \left( \dfrac{n}{n^2}+\dfrac{n}{n^2+1^2}+\dfrac{n}{n^2+2^2}+ \cdots + \dfrac{n}{n^2+(n-1)^2} +\dfrac{n}{n^2+n^2} \right)$