Im just not able to wrap my head around how to find this limit:
lim [arctan(x)] -x
x-> 0 --------------
x^3
What level of math are you at? Do you know L'hopital's?
You could also use maclaurin series
$\displaystyle \begin{aligned}\lim_{x\to{0}}\frac{\arctan(x)-x}{x^3}&=\lim_{x\to{0}}\frac{x-\frac{x^3}{3}+\cdots-x}{x^3}\\
&=\lim_{x\to{0}}\frac{\frac{-x^3}{3}+\cdots}{x^3}\\
&=\frac{-1}{3}\end{aligned}$