Finding a limit including an improper integral: $\lim_{n\to\infty}⁡ \int_0^{\sqrt n} \left({1-\frac{x^2}{n}}\right)^n\,dx$
Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$