How to solve this complex limits at infinity with trig? Please consider this limit question
$$\lim_{x\rightarrow\infty}\frac{a\sin\left(\frac{a(x+1)}{2x}\right)\cdot \sin\left(\frac{a}{2}\right)}{x\cdot \sin\left(\frac{a}{2x}\right)}$$
How should I solve this? I have no idea where to start please help.

Trivial with equivalents: 
as $x\to\infty$,


*

*$\dfrac{a(x+1)}{2x}\to \dfrac a2$, so $\sin\dfrac{a(x+1)}{2x}\sim_\infty \sin \dfrac a2$,

*$\sin\dfrac{a}{2x}\sim_\infty\dfrac{a}{2x} $,
so that
$$\frac{a\sin \dfrac{a(x+1)}{2x}\,\sin\dfrac{a}{2}}{x\cdot \sin \dfrac{a}{2x}}\sim_\infty\frac{a\sin^2\dfrac a2}{x\,\dfrac a{2x}}=2\sin^2\dfrac a2=1-\cos a.$$

