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?
$$\lim_{x \to \infty}x^{2}\ln\left(\cos\left(\pi \over x\right)\right)$$ 

Let $x=1/t$. So we are interested in the limit as $t\to 0^+$ of $\ln(\cos(\pi t))/t^2$.
One round of L'Hospital's Rule brings us to 
$$\lim_{t\to 0^+} -\frac{\pi}{2\cos(\pi t)} \frac{\sin(\pi t)}{t}.$$
The first part is nicely behaved near $0$. For $\lim_{t\to 0^+} \frac{\sin(\pi t)}{t}$, use L'Hospital's Rule, or simpler tools.
