Evaluate: $\lim_{n\to\infty} \int_a^{\infty}\frac{n^2xe^{-n^2x^2}}{1+x^2}\,dx.$
Solution 1:
Hint
Another way is the squeeze theorem:
$$ \int_a^{\infty}\frac{n^2xe^{-n^2x^2}}{1+x^2}\,dx \le \int_{a}^{\infty}n^2xe^{-n^2x^2}\,dx=\frac{1}{2}\exp(-n^2a^2). $$