TITLE: A limit with Product and Cosine
QUESTION [5 upvotes]: Can anyone help me please to find the limit of this series:
$$\lim_{x \to 0}\frac{\displaystyle\prod_{k=1}^{n}\left(1-\sqrt[k]{\cos x}\right)}{x^{2n-2}}$$
Thanks for all :D

REPLY [3 votes]: Note that by elementary limits
$$\lim_{x \to 0}\frac{1-\sqrt[k]{\cos x}}{x^2}=\lim_{x \to 0}\frac{1-\sqrt[k]{\cos x}}{\ln \sqrt[k]{\cos x}}\times \frac{1}{k}\lim_{x \to 0}\frac{\ln \cos x}{x^2}=-1\times \left(-\frac{1}{2k}\right)=\frac{1}{2k}$$
and then 
$$\lim_{x \to 0}\frac{\prod_{k=1}^{k=n}1-\sqrt[k]{\cos x}}{x^{2n}}=\prod_{k=1}^{k=n}\frac{1}{2k}=\frac{1}{2^n n!}$$
Chris.