A limit with an arctan I need to calculate another limit, $$ \underset{x\rightarrow\infty}{\lim}\left(\frac{2}{\pi}\arctan x\right)^{\sin\left(2/x\right)}$$
I have no idea how to start or what to do because I've never dealt with an arctan in a limit before.
What should I do first?

First of all: don't panic! :-)
There is a nice identity for the arctangent:
$$
\arctan x+\arctan\frac{1}{x}=
\begin{cases} \pi/2 & x>0 \\[4px] -\pi/2 & x<0 \end{cases}
$$
Since the limit is for $x\to\infty$, it's not restrictive to assume $x>0$, so
$$
\arctan x=\frac{\pi}{2}-\arctan\frac{1}{x}
$$
and, setting $y=1/x$, the limit becomes
$$
\lim_{y\to0^+}\Bigl(1-\frac{2}{\pi}\arctan y\Bigr)^{\sin2y}
$$
which is not an indeterminate form because it is $1^0=1$.
