Question

$\underset{x\to 0}{\lim }\frac{sin(\pi co{s}^{2}x)}{x^2}$ is equal to

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

Answer 1

Answer: (D)

$\underset{x\to 0}{\lim }\frac{\sin \left(\pi {\cos }^{2}x\right)}{x^2}$
$=\underset{x\to 0}{\lim }\frac{\sin (\pi \left(1-{\sin }^{2}x\right)}{x^2}$
$=\underset{x\to 0}{\lim }\frac{\sin \left(\pi -\pi {\sin }^{2}x\right)}{x^2}$
$=\underset{x\to 0}{\lim }\frac{\sin \left(\pi {\sin }^{2}x\right)}{x^2}[\because \sin (\pi -\theta )=\sin \theta ]$
$=\underset{x\to 0}{\lim }\frac{\sin \left(\pi {\sin }^{2}x\right)}{\left(\pi {\sin }^{2}x\right)}\times \frac{\pi {\sin }^{2}x}{x^2}$
$=\underset{x\to 0}{\lim }1\times \pi {\left(\frac{\sin x}{x}\right)}^2=\pi$