Find the limit

Let $\mathfrak T$ denote the given limit. $\Large \mathfrak {T}=\lim_{n \to \infty}\sum_{k=1}^{n}\frac{n}{k^2+n^2}\\\Large=\lim_{n \to \infty}\dfrac 1n\sum_{k=1}^{n}\dfrac{1}{\left(\color{#D61F06}{\dfrac{k}{n}}\right)^2+1}$

By Reimann Sums ,

$\Large\mathfrak T=\displaystyle\int_0^1\dfrac{\mathrm{d}x}{\color{#D61F06}{x^2}+1}$ $\Large=\left|\tan^{-1}x\right|_0^{1}=\tan^{-1}(1)$ $\Large=\pi/4\approx \boxed{0.785398}$