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

Calculate

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

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Where did this problem come from? What have you tried? –  robjohn Feb 4 '13 at 17:15

\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.

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why the switch integral/limit works ? I mean does it work all the time ? –  aziiri Feb 4 '13 at 16:47
@aziiri: I added some commentary regarding the change of variables and the limit. Let me know if it is still unclear. –  robjohn Feb 4 '13 at 16:54