Question

Find
$\underset{x\to a}{lim}\frac{\sin x-\sin a}{x^2-a^2}$

Answer 1

We have,

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

Apply L-Hospital rule,

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

${lim}_{x\to 0}\left(\frac{\cos x}{2x}\right)$

Apply L-Hospital rule,

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

$=\frac{-\sin 0}{2}$

$=0$

Hence, this is the answer.