Question

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

Answer 1

$\begin{array}{c}{lim}_{x\to a}\frac{\sin x-\sin a}{\left(x-a\right)}\\ {lim}_{x\to a}\frac{\sin \left(x-a+a\right)-\sin a}{\left(x-a\right)}\\ {lim}_{x\to a}\frac{\sin \left(x-a\right)\cdot \cos a}{\left(x-a\right)}+\frac{\cos \left(x-a\right)\cdot \sin a}{\left(x-a\right)}-\frac{\sin a}{\left(x-a\right)}\\ {lim}_{x\to a}\frac{\sin \left(x-a\right)}{\left(x-a\right)}\cdot \cos a+{lim}_{\left(x-a\right)\to a}\left\{\frac{\cos \left(x-a\right)\cdot \sin a-\sin a}{\left(x-a\right)}\right\}\\ 1\times \cos a+0\left[b{lim}_{x\to a}\frac{\sin x}{x}=1\right]\\ =\cos a\end{array}$