TITLE: $ \lim_{n\to\infty}\int_{0}^{\infty}\frac{e^{-x}\cos(x)}{\frac{1}{n}+nx^2}dx $
QUESTION [5 upvotes]: Calculate 
$\displaystyle \lim_{n\to\infty}\int_{0}^{\infty}\frac{e^{-x}\cos(x)}{\frac{1}{n}+nx^2}dx $

REPLY [5 votes]: $$
\begin{align}
\lim_{n\to\infty}\int_0^\infty\frac{e^{-x}\cos(x)}{\frac1n+nx^2}\,\mathrm{d}x
&=\lim_{n\to\infty}\int_0^\infty\frac{e^{-x}\cos(x)}{1+n^2x^2}\,\mathrm{d}nx\\
&=\lim_{n\to\infty}\int_0^\infty\frac{e^{-u/n}\cos(u/n)}{1+u^2}\,\mathrm{d}u\\
&=\int_0^\infty\frac{\mathrm{d}u}{1+u^2}\tag{$\ast$}\\[3pt]
&=\frac\pi2
\end{align}
$$
where $u=nx$ and $(\ast)$ is by Dominated Convergence.