\displaystyle \lim_{x\rightarrow 0}(\frac{\arctan x}{x})^{\frac{1}{x^2}}=\lim_{x\rightarrow 0}exp{\frac{1}{x^2}\ln(\frac{\arctan x}{x})}=\lim_{x\rightarrow 0} e^{-\frac{1}{3}+\mathcal{O}_0(x^6)}=e^{-\frac{1}{3}}