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?

$\lim_{x \to \pi/2} \tan x = \infty$, so $\lim_{x \to \infty} \arctan x = \frac{\pi}{2}$, as $\tan x$ and $\arctan x$ are inverse functions of each other for $x \in [0, \pi/2)$.
The rest is just direct substitution. You should get:

 $$\lim_{x \to \infty} \left(\frac{2}{\pi} \cdot \frac{\pi}{2} \right)^{\sin(0)} = \lim_{x \to \infty} 1^0 = \boxed{1}.$$

