91Ó°ÊÓ

Compute the numbers \(e^{i \pi / 2}, e^{-i \pi / 4}, e^{5 \pi i / 6}, e^{\ln 2-i \pi / 6}\)

Let \(\varphi\) be a \(C^{1}\) function such that \(\varphi\) and \(\varphi^{\prime}\) are bounded on the real axis. Compute the limit $$ \lim _{n \rightarrow \infty} \frac{2 n^{3}}{\pi} \int_{-\infty}^{\infty} \frac{x}{\left(1+n^{2} x^{2}\right)^{2}} \varphi(x) d x $$