$\lim_{x\rightarrow +\infty}x^{2}\left ( \frac{e^{\frac{1}{x^{2}}}-\cos \frac{1}{x}}{\arctan x} \right )$
$\begin{align*} \lim_{x\to+\infty}x^{2}\cdot \left ( \dfrac{e^{\tfrac{1}{x^{2}}}-\cos\frac{1}{x}}{\arctan x} \right )&=\lim_{x\to+\infty}\left (x^2\cdot \dfrac{e^{\tfrac{1}{x^{2}}}-1+1-\cos\frac{1}{x}}{\pi/2} \right )\\&=\lim_{x\to+\infty}\left (x^2\cdot\frac{1/{x^2}+\frac12\cdot 1/{x^2}}{\pi/2} \right )\\&=\frac3\pi \end{align*}$