I'm trying to understand how this problem works:
$$\lim_{x \rightarrow +\infty} \frac{\int_0^{x} (arctg(t))^2 ~ dt}{\sqrt{x^2+1}} = [\frac{\infty}{\infty}] = \lim_{x \rightarrow +\infty} \frac{ (arctg(t))^2\Bigg|_{0}^{x}}{\frac{2x}{2\sqrt{x^2+1}}} = \lim_{x \rightarrow +\infty} \frac{ (arctg(t))^2\Bigg|_{0}^{x}}{\frac{x}{x\sqrt{1+\frac{1}{x^2}}}} = \lim_{x \rightarrow +\infty} \frac{ (arctg(t))^2\Bigg|_{0}^{x}}{\frac{1}{\sqrt{1+\frac{1}{x^2}}}} = \frac{\pi^2}{4} $$
Specifcally the parts about finding the limit of the integral and then computing its derivative, everything else makes sense to me. I know it's a simple problem, but I don't fully understand how it works. Could someone help out?