How can I calculate $\displaystyle\lim_{x \to \infty}x^{2}\ln\left(\cos \left(\pi/x\right)\right) $? Does anybody know how to solve this?
$$\lim_{x \to \infty}x^{2}\ln\left(\cos\left(\pi \over x\right)\right)$$ 

Hint: Use a change of variables $u = \frac{\pi}{x}$ and then it will be solvable by first year calculus (you do not even have to use a Taylor's series).
