Question

Evaluate: $\underset{x\to 0}{lim}\frac{\sin (1+x)-\sin (1-x)}{x}$

• A. $2$
• B. $2\cos 1$
• C. $\cos 2$
• D. $0$

Answer 1

The correct option is B $2\cos 1$

$\underset{x\to 0}{lim}\frac{\sin (1+x)-\sin (1-x)}{x}$

$=\underset{x\to 0}{lim}\frac{2\sin \left(\frac{1+x-(1-x)}{2}\right)\cos \left(\frac{(1+x)+(1-x)}{2}\right)}{x}$

$=\underset{x\to 0}{lim}\frac{2\sin x\cos 1}{x}$

$=2\cos 1$ $(\because {lim}_{x\to 0}\frac{\sin x}{x}=1)$