Question

Найти предел
$lim_{n to infty}cos^n frac{x}{sqrt{n}}$

Answer 1

$displaystyle lim_{n to infty} bigg(cos frac{x}{sqrt{n}} bigg)^n=big{1^{infty}big}=lim_{n to infty}bigg(1+cosfrac{x}{sqrt{n}}-1bigg)^big{ncdotfrac{cosfrac{x}{sqrt{n}}-1}{cosfrac{x}{sqrt{n}}-1}}=\ \ \ =e^bigg{displaystyle lim_{n to infty}nbigg(cosfrac{x}{sqrt{n}}-1bigg)}=e^bigg{displaystylelim_{n to infty}ncdot bigg(-2sin^2frac{x}{2sqrt{n}}bigg)}=e^bigg{displaystyle -lim_{n to infty}2ncdotbigg(frac{x}{2sqrt{n}}bigg)^2}=\ \ \ =e^bigg{displaystylelim_{n to infty}2ncdotfrac{x^2}{4n}}=e^bigg{-x^2/2}$