Question

Solve : $\underset{x\to 0}{lim}\frac{\sin x\sin (\frac{\pi }{3}+x)\sin (\frac{\pi }{3}-x)}{x}$

• A. $\frac{3}{4}$
• B. $\frac{1}{4}$
• C. $\frac{4}{3}$
• D. $0$

Answer 1

The correct option is A $\frac{3}{4}$

Solve : $\underset{x\to 0}{lim}\frac{\sin x\sin (\frac{\pi }{3}+x)\sin (\frac{\pi }{3}-x)}{x}$
$=\underset{x\to 0}{lim}\frac{\sin x}{x}\times \underset{x\to 0}{lim}\sin (\frac{\pi }{3}+x)\times \underset{x\to 0}{lim}\sin (\frac{\pi }{3}-x)$
$=1\times \sin \frac{\pi }{3}\times \sin \frac{\pi }{3}$
$=1\times \frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}$
$=\frac{3}{4}$