# Convergence of $\lim\limits_{n\to\infty}\frac{1}{n}\int_\limits{-n}^{n}\frac{1-e^{-nx^2}}{x^2(1+nx^2)}dx$

Prove the convergence of the following integral: $$\lim\limits_{n\to\infty}\frac{1}{n}\int_\limits{-n}^{n}\frac{1-e^{-nx^2}}{x^2(1+nx^2)}dx$$

I tried to use the obvious Weierstrass comparison test in the following way:

$$\lim\limits_{n\to\infty}\frac{1}{n}\int_\limits{-n}^{n}\frac{1-e^{-nx^2}}{x^2(1+nx^2)}dx \leqslant \lim\limits_{n\to\infty}\frac{1}{n}\int_\limits{-n}^{n}\frac{1-e^{-nx^2}}{x^2}dx = \lim\limits_{n\to\infty}\int_\limits{-n}^{n}\frac{1-e^{-nx^2}}{nx^2}dx$$ However after thinking about a variable substitution I concluded that it would complicate the expression.

Question:

How should I prove the integral converges? What are your suggestions?