$ \lim_{n\to\infty}\int_{0}^{\infty}\frac{e^{-x}\cos(x)}{\frac{1}{n}+nx^2}dx $

Question

Calcular

$\displaystyle \lim_{n\to\infty}\int_{0}^{\infty}\frac{e^{-x}\cos(x)}{\frac{1}{n}+nx^2}dx $

Answer 1

$$ \begin{align} \lim_{n\to\infty}\int_0^\infty\frac{e^{-x}\cos(x)}{\frac1n+nx^2}\,\mathrm{d}x &=\lim_{n\to\infty}\int_0^\infty\frac{e^{-x}\cos(x)}{1+n^2x^2}\,\mathrm{d}nx\\ &=\lim_{n\to\infty}\int_0^\infty\frac{e^{-u/n}\cos(u/n)}{1+u^2}\,\mathrm{d}u\\ &=\int_0^\infty\frac{\mathrm{d}u}{1+u^2}\tag{%#%#%}\\[3pt] &=\frac\pi2 \end {align} $$ donde$\ast$ y$u=nx$ es por convergencia dominada.