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.
$$\lim_{x\to 0} \cos \bigg(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \bigg)=\lim_{x\to 0} \cos \bigg(\pi x^2 (\frac {\cos (6x)} {\sin (\frac {x} {2}) \sin 6x}  \bigg)$$
 A: HINT: 
$$\begin{align*}
\lim_{x\to 0}~\cos\left(\pi x^2 \csc \left(\frac {x} {2}\right) \cot (6x)\right)&=\lim_{x\to 0}~\cos\left(\frac{\pi x^2}{\sin(x/2)\tan 6x}\right)\\
&=\lim_{x\to 0}~\cos\frac{\pi}3\left(\frac{x/2}{\sin(x/2)}\cdot\frac{6x}{\tan 6x} \right)\\
&=\cos\frac{\pi}3\left(\lim_{x\to 0}\frac{x/2}{\sin(x/2)}\right)\left(\lim_{x\to 0}\frac{6x}{\tan 6x}\right)\;,
\end{align*}$$
since the cosine is a continuous function. The limits in the last line are ones that you should know.
A: $$\pi x^2\csc\left(\frac{x}{2}\right)\cot(6x)=\pi x^2\cdot\frac{2}{x}\cdot\frac{1}{6x}+\ldots=\frac{\pi}{3}+O(x)$$
so the limit is $\cos(\pi/3)=\frac{1}{2}$.
A: Hints: express all the trig functions in terms of the sine and the cosine, and then manipulate things so you can use what you might know about $\lim_{x\to0}((\sin x)/x)$. 
A: 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
$$ \lim_{x\to 0}\cos \Big(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \Big)
=\cos \Big(\lim_{x\to 0}\pi x^2 \csc (\frac {x} {2}) \cot (6x) \Big)=\cos\frac{\pi}{3}=\frac{1}{2}.$$
A: $$\begin{align*}
\lim_{x\to 0} \cos (\pi x^2 \csc (\frac {x} {2}) \cot (6x))
&=\cos(\lim_{x\to 0}\pi x^2 \csc (\frac {x} {2}) \cot (6x))\\
&=\cos(\lim_{x\to 0}\frac{\pi}{3}.\frac{\frac{x}{2}}{\sin\frac{x}{2}}.\frac{6x}{\sin6x}.\cos6x)\\
&=\cos(\frac{\pi}{3}.\lim_{x\to 0}\frac{\frac{x}{2}}{\sin\frac{x}{2}}.\lim_{x\to 0}\frac{6x}{\sin6x}.\lim_{x\to 0}\cos6x)\\
&=\cos\frac{\pi}{3}=\frac{1}{2}
\end{align*}$$
