\lim_{x\rightarrow\infty}\frac{e^{1/x}-1}{\pi-2\arctan(x)}=\lim_{x\rightarrow\infty}\frac{e^{1/x}}{2x^2/(1+x^2)}=\lim_{x\rightarrow\infty}\frac{e^{1/x}}{2/(1+1/x^2)}=\frac 12