Evaluate the following limits:

$=\lim _{x \rightarrow 0} \frac{[\sin (2+x)-\sin (2-x)]}{x}$

$=\lim _{x \rightarrow 0} \frac{\left[2 \times \cos \frac{(2+x+2-x)}{2} \times \sin \frac{(2+x-2+x)}{2}\right]}{x}$

$=\lim _{x \rightarrow 0} \frac{(2 \times \cos 2 \times \sin x)}{x}$

$=2 \cos 2 \lim _{x \rightarrow 0} \frac{\sin x}{x}$

$=2 \cos 2$

$\lim _{x \rightarrow 0} \frac{[\sin (2+x)-\sin (2-x)]}{x}=2 \cos 2$