Question

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

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

Answer 1

The correct option is D $\pi$

$li{m}_{x\to 0}\frac{\sin \left(\pi {\cos }^{2}x\right)}{x^2}=li{m}_{x\to 0}\frac{\sin \left(\pi \right(1-{\sin }^{2}x\left)\right)}{x^2}$

$=li{m}_{x\to 0}\frac{\sin (\pi -\pi {\sin }^{2}x)}{x^2}=li{m}_{x\to 0}\frac{\sin \left(\pi {\sin }^{2}x\right)}{x^2}=li{m}_{x\to 0}\frac{\pi {\sin }^{2}x}{x^2}$ [since $\pi {\sin }^{2}x$ tends to $0$]

$=li{m}_{x\to 0}\pi \times (\frac{\sin x}{x}{)}^2=\pi$