Lim(n→∞)(n-√(n^2+3)); б) lim/(n→∞) 7^n/n!

$\lim_{n \to \infty} (n- \sqrt{n^2+3})= \lim_{n \to \infty} \frac{ (n- \sqrt{n^2+3})\cdot (n+ \sqrt{n^2+3})}{(n+ \sqrt{n^2+3})}= \\ \\ = \lim_{n \to \infty} \frac{ (n)^2- (\sqrt{n^2+3})^2}{(n+ \sqrt{n^2+3})}= \lim_{n \to \infty} \frac{ n^2- n^2-3}{n+ \sqrt{n^2+3}}= \lim_{n \to \infty} \frac{ -3}{n+ \sqrt{n^2+3}}=0$

$\lim_{n \to \infty} \frac{7^{n}}{n!} = \lim_{n \to \infty} \frac{7^{n}}{ \sqrt{2 \pi n}\cdot ( \frac{n}{e})^{n} } = \\ \\ = \lim_{n \to \infty} \frac{(7e)^{n}}{ \sqrt{2 \pi n}\cdot (n)^{n} } = \lim_{n \to \infty} \frac{1}{ \sqrt{2 \pi n} } \cdot (\frac{7e}{n})^n =0$

Формула Стирлинга

n! ≈ $\sqrt{2 \pi n} \cdot ( \frac{n}{e})^{n}$