Let $f\left(x\right)=\left\{\begin{array}{cc}\frac{x}{\sin x},& x\in (0,1)\\ 1,& x=0\end{array},\right.$ consider the integral $I_n=\sqrt{n}{\int }_0^{1/n}f\left(x\right)e^{-nx}dx$. Then $\underset{n\to \infty }{\lim }{I}_n$

Let $\mathbb{N}$ be the set of natural numbers. For $n\in \mathrm{\mathbb{N}},$ define $I_n={\int }_0^{\pi }\frac{x{\sin }^{2n}\left(x\right)}{{\sin }^{2n}\left(x\right)+{\cos }^{2n}\left(x\right)}dx.$

Then for $m,n\in \mathrm{\mathbb{N}}$