A calculus problem by Jose Sacramento

Relevant wiki: Riemann Sums

$\begin{aligned} L & = \lim_{n \to \infty} \left( \frac n{n^2+1} + \frac n{n^2+4} + \frac n{n^2+9} + \cdots + \frac n{n^2+n^2} \right) \\ & = \lim_{n \to \infty} \sum_{k=1}^n \frac n{n^2+k^2} & \small \color{#3D99F6}{\text{Divide up and down with }n^2} \\ & = \lim_{n \to \infty} \sum_{k=1}^n \frac {\frac 1n}{1+\frac {k^2}{n^2}} \\ & = \lim_{n \to \infty} \frac 1n \sum_{k=1}^n \frac 1{1+\frac {k^2}{n^2}} & \small \color{#3D99F6}{\text{By Riemann sums}} \\ & = \int_{a=0}^{b=1} \frac 1{1+x^2} dx & \small \color{#3D99F6}{a = \lim_{n \to \infty} \frac kn = 0, \ b = \lim_{n \to \infty} \frac nn = 1} \\ & = \tan^{-1} x \ \bigg|_0^1 = \frac \pi 4 \approx \boxed{0.785} \end{aligned}$