Question

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

• A. $\frac{\pi }{2}$
• B. $1$
• C. $-\pi$
• D. $\pi$

Answer 1

The correct option is D $\pi$

$\underset{x\to 0}{lim}\frac{\sin \left(\pi {\cos }^{2}x\right)}{x^2}=\underset{x\to 0}{lim}\frac{\sin (\pi -\pi {\sin }^{2}x)}{x^2}$
$=\underset{x\to 0}{lim}\frac{\sin \left(\pi {\sin }^{2}x\right)}{x^2}$
$=\underset{x\to 0}{lim}[\frac{\sin \left(\pi {\sin }^{2}x\right)}{\pi {\sin }^{2}x}\cdot \frac{\pi {\sin }^{2}x}{x^2}]$
$=\pi$