<s>[INST] Calculate a hard limit Calculate: $$
\lim_{x\to+\infty} x\left( \frac{1}{x^2+1^2}+\frac{1}{x^2+2^2}+\dots+\frac{1}{x^2+x^2}\right)$$
 [/INST] $$
\begin{align}
& \phantom{=} \lim_{x\to+\infty} x\left( \frac{1}{x^2+1^2} + \frac{1}{x^2+2^2} + \dots + \frac{1}{x^2+x^2}\right) \\[8pt]
& = \lim_{x\to\infty} \frac 1 x \left( \frac{1}{1+\frac{1^2}{x^2}} + \frac{1}{1+\frac{2^2}{x^2}} +\cdots+\frac1{1+\frac{x^2}{x^2}} \right) \\[8pt]
& = \int_0^1 \frac{1}{1+w^2} \, dw = \frac \pi 4.
\end{align}
$$
</s>