$\underset{x\to 0}{lim} \frac{\sin \left(\pi {\cos }^{2}x\right)}{x^2}$ equals

Solution:

We have,

$\underset{x\to 0}{lim} \frac{\sin \left(\pi {\cos }^{2}x\right)}{x^2}$

$\begin{array}{l}=\underset{x\to 0}{lim} \frac{\sin \left(\pi -\pi {\cos }^{2}x\right)}{x^2}\frac{x}{x^2}\\ =\underset{x\to 0}{lim} \frac{\sin \left(\pi {\sin }^{2}x\right)}{\pi {\sin }^{2}x}\times \frac{\pi {\sin }^{2}x}{x^2}=1\times \pi \times (1{)}^2=\pi \end{array}$