TITLE: Evaluate $\lim_{h \to 0} \frac{\sin(\frac{h}{2})-\frac{h}{2}}{h\sin(\frac{h}{2})}$ without l'Hospital
QUESTION [1 upvotes]: $$
\lim_{h \to 0} \frac{\sin(\frac{h}{2})-\frac{h}{2}}{h\sin(\frac{h}{2})}
$$
I've worked the last few hours on this equation and still didnt find a way to evaluate it.
The idea I had was to bound this expression like this:
$$
? \leqslant \lim_{h \to 0} \frac{\sin(\frac{h}{2})-\frac{h}{2}}{h\sin(\frac{h}{2})}\leqslant0
$$
If you could guide me maybe to find this expression or guide me on another track.
Thanks in advance !

REPLY [1 votes]: This question can be solved by using the expansion of the function $\sin x$ which is as $$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+  \cdots  $$
Thus substituting $\frac{h}{2}$ in place of $x$ and then solving accordingly will give the answer.
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