Question

$\begin{array}{c}lim\\ x\to 0\end{array}\frac{\sin (1+x)-\sin (1-x)}{x}$

Answer 1

We have,

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

Applying L’ Hospital rule

$={lim}_{x\to 0}\frac{\cos (1+x)-\cos (1-x)(-1)}{1}$

$={lim}_{x\to 0}\frac{\cos (1+x)+\cos (1-x)}{1}$

Taking limit and we get,

$=\cos 1+\cos 1$

$=2\cos 1$

Hence, this is the answer.