Q:

calculation of limit of $\lim_{\theta\to0}\frac{\cos\left(\frac\pi2\cos\theta\right)}{\sin\theta}$

I try with taylor expansion to prove that
$$\lim_{\theta\to0}\frac{\cos\left(\frac\pi2\cos\theta\right)}{\sin\theta}=0$$
but I have always a singularity when $\sin\theta\approx\theta$
Any help is welcome

A:

Taylor expansion gives you
$$\cos\left(\frac{\pi}{2}\cos\theta\right)\approx 
\cos\left(\frac{\pi}{2}(1-\theta^2/2)\right)\approx
\cos\left(\frac{\pi}{2}-\frac{\pi}{4}\theta^2\right)=\sin\left(\frac{\pi}{4}\theta^2\right)\approx \frac{\pi}{4}\theta^2
$$
while the denominator is just $\sin\theta\approx \theta$. You see that the numerator has a higher leading order and thus overpowers the denominator, resulting in zero limit.

