Question

Solve:
$\underset{x\to }{lim}\frac{\sin (2+x)-\sin (2-x)}{x}$

Answer 1

We have,

${lim}_{x\to 0}\frac{\sin \left(2+x\right)-\sin \left(2-x\right)}{x}$

${lim}_{x\to 0}\frac{\left(\sin 2.\cos x+\cos 2.\sin x-\sin 2.\cos x+\cos 2.\sin x\right)}{x}{lim}_{x\to 0}\frac{2\sin x.\cos 2}{x}\left[{lim}_{\theta \to 0}\frac{\sin \theta }{\theta }=1\right]$

$=2\times \cos 2=2\cos 2$

Hence, this is the answer.