Question

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

Answer 1

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

$$\begin{gathered} \Rightarrow \underset{x\to 0}{\lim }\left[\frac{2\cos \left({\displaystyle \frac{2+x+2-x}{2}}\right)\sin \left({\displaystyle \frac{2+x-2+x}{2}}\right)}{x}\right] \\ \Rightarrow \underset{x\to 0}{\lim }\left[\frac{2\cos 2\sin x}{x}\right] \\ \Rightarrow 2\cos 2\left[\because \underset{x\to 0}{\lim }\frac{\sin x}{x}=1\right] \end{gathered}$$