Find  $\lim_{x\to 0} \cos \big(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \big) $ Find the limit $$\lim_{x\to 0} \cos \bigg(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \bigg)$$
I dont even know where to get started... Some hints and solutions would be appreciated! Thanks in advance!
P.S typed this on an iphone, sorry for any mistakes will edit soon.
EDIT 
Here are my current workings.
$$\lim_{x\to 0} \cos \bigg(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \bigg)=\lim_{x\to 0} \cos \bigg(\pi x^2 (\frac {\cos (6x)} {\sin (\frac {x} {2}) \sin 6x}  \bigg)$$

Note that 
$$\pi x^2 \csc (\frac {x} {2}) \cot (6x)=\frac{\pi}{3}\cdot\frac{\frac{x}{2}}{\sin(\frac{x}{2})}\cdot\frac{6x}{\sin(6x)}\cos(6x).$$
Since $\lim_{x\to 0}\frac{\sin x}{x}=1$ and $\lim_{x\to 0}\cos(x)=1$, we have 
$$\lim_{x\to 0}x^2 \csc (\frac {x} {2}) \cot (6x)=\frac{1}{3}.$$
Since $\cos$ is a continuous function, we have
$$ \lim_{x\to 0}\cos \Big(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \Big)
=\cos \Big(\lim_{x\to 0}\pi x^2 \csc (\frac {x} {2}) \cot (6x) \Big)=\cos\frac{\pi}{3}=\frac{1}{2}.$$
