Question

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

Answer 1

We have,

${lim}_{x\to 0}\frac{\sin x.\sin \left(\frac{\pi }{3}+x\right)\sin \left(\frac{\pi }{3}-x\right)}{x}$

${lim}_{x\to 0}\frac{\sin x.\sin \left(\frac{\pi }{3}+0\right)\sin \left(\frac{\pi }{3}-0\right)}{x}$

${lim}_{x\to 0}\frac{\sin x.\sin \left(\frac{\pi }{3}\right)\sin \left(\frac{\pi }{3}\right)}{x}$

$={\left(\frac{\sqrt{3}}{2}\right)}^2{lim}_{x\to 0}\frac{\sin x}{x}$

$=\frac{3}{4}\times 1$

$=\frac{3}{4}$

Hence, this is the answer.