Question

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

Answer 1

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

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

Формула Стирлинга

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

Answer 2

вау, спасибо