Question

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

Answer 1

We have,

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

$\underset{x\to 0}{lim}\frac{\sin \left(\pi \right(1-{\sin }^{2}x\left)\right)}{x^2}$

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

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

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

We know that

${lim}_{x\to 0}\frac{\sin x}{x}=1$

Therefore,

$=1\times \pi \times 1$

$=\pi$

Hence, this is the answer.