how to solve this
$$\lim_{\frac{sin(a+x) - sin(a-x) }{x}}
x\Rightarrow0$$
Expand sin(a+x) as a series in x:
sin(a+x) = sin(a) + x*cos(a) + higher order terms involving multiples of x.
Similarly:
sin(a-x) = sin(x) - x*cos(a) + higher order terms
Therefore, sin(a+x) - sin(a-x) = 2x*cos(a) + higher order terms
(sin(a+x) - sin(a-x))/x = 2cos(a) + higher order terms.
The higher order terms all contain multiples of x, so when x goes to zero, these go to zero and we are left with:
$$\lim_{x \rightarrow 0}\frac{\sin(a+x)-\sin(a-x)}{x}=2\cos(a)$$