Question

Evaluate
$\underset{x\to 0}{lim}\frac{\sin \left(3+x\right)-\sin \left(3-x\right)}{x}$

• A. $-2\cos 3$
• B. $2\cos 3$
• C. $\cos 3$
• D. None of these

Answer 1

Answer: (B)

$2\cos 3$

$\underset{x\to 0}{lim}\frac{\sin \left(3+x\right)-\sin \left(3-x\right)}{x}$

We can observe that it is of the form $\frac{0}{0}$

So, We will apply L Hospital rule

$\underset{x\to 0}{lim}\frac{\frac{d}{dx}[\sin \left(3+x\right)-\sin \left(3-x\right)]}{\frac{d}{dx}x}$

$\Rightarrow \underset{x\to 0}{lim}\frac{\cos \left(3+x\right)+\cos \left(3-x\right)}{1}=\frac{2\cos 3}{1}=2\cos 3$