calculus - Limit involving inverse tan function

I solved the following limit using L'Hospital's rule, but can't seem to solve it without using L'Hospital's.
$$\lim_{x\to\infty} \frac{e^{-1/x^2}-1}{2\arctan x-\pi}$$

I would like a hint as to how to get started.

I was also wondering how to approach inverse trigonometric functions in general when they appear in limits, since I didn't understand any solutions to this type of problem that I looked up.