Question

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

Answer 1

$\displaystyle \lim_{n \to \infty} \bigg(\cos \frac{x}{\sqrt{n}} \bigg)^n=\big\{1^{\infty}\big\}=\lim_{n \to \infty}\bigg(1+\cos\frac{x}{\sqrt{n}}-1\bigg)^\big{n\cdot\frac{\cos\frac{x}{\sqrt{n}}-1}{\cos\frac{x}{\sqrt{n}}-1}}=\\ \\ \\ =e^\bigg{\displaystyle \lim_{n \to \infty}n\bigg(\cos\frac{x}{\sqrt{n}}-1\bigg)}=e^\bigg{\displaystyle\lim_{n \to \infty}n\cdot \bigg(-2\sin^2\frac{x}{2\sqrt{n}}\bigg)}=e^\bigg{\displaystyle -\lim_{n \to \infty}2n\cdot\bigg(\frac{x}{2\sqrt{n}}\bigg)^2}=\\ \\ \\ =e^\bigg{\displaystyle\lim_{n \to \infty}2n\cdot\frac{x^2}{4n}}=e^\bigg{-x^2/2}$