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How to compute this limit:

$$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$

Please give me some hint.

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closed as off-topic by choco_addicted, Chinnapparaj R, A. Pongrácz, RRL, José Carlos Santos Dec 5 '18 at 14:35

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$$\begin{align}\lim_{n\to\infty}\sum_{1\le r\le n}\frac{n}{n^2+r^2} &=\lim_{n\to\infty}\frac1n\sum_{1\le r\le n}\frac1{1+\left(\frac rn\right)^2}\\ &=\int_0^1\frac{dx}{1+x^2}\end{align}$$

$$\text{as }\lim_{n \to \infty} \frac1n\sum_{r=1}^n f\left(\frac rn\right)=\int_0^1f(x)dx$$

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