Originally Posted by

**davidmccormick** Here is the question:

Evaluate:

$\displaystyle

\lim_{n\to\infty} \int_0^{\infty} (1 + {\frac{x}{n}})^{-n} sin{\frac{x}{n}} dx

$

$\displaystyle

\lim_{n\to\infty} \int_0^{1} (1+nx^{2})(1+x^{2})^{-n}dx

$

For the first integral I use the DCT, noting that $\displaystyle (1 + {\frac{x}{n}})^{-n} $ converges to $\displaystyle e^{-x}$ and that $\displaystyle sin$ is bounded by 1. Then taking the limit inside, the integrand converges to 0 and therefore the upper integral is equal to 0. Is this correct? Any hints on the second one would be appreciated.