Evaluating $ \lim_{n\to\infty}\sum_{i=0}^n\frac{n}{n^2+i^2} $
$$ \sum_{i=0}^n\frac{1}{n^2+i^2} = \frac 1 n \sum_{i=0}^n \frac 1 {1 + \left( \frac i n \right)^2} \cdot \frac 1 n = \frac 1 n \sum_{i=0}^n \arctan'\left( \tfrac i n \right)\cdot \tfrac 1 n $$
That last sum is a Riemann sum that approximates $\displaystyle\int_0^1 \arctan'(x)\,dx = \arctan 1 - \arctan 0 = \frac \pi 4,$ but outside the sum there is a factor of $1/n$ that approaches $0,$ so ultimately the limit is $0.$