TITLE: Examples of Calculations using Lebesgue Dominated Convergence
QUESTION [7 upvotes]: I came across the following problem in my self-study, and wanted to know how to use Lebesgue Dominated Convergence to compute any of the following limits:
(a) $\lim\limits_{n \rightarrow \infty}$ $\int_0^\infty$ $(1+(x/n))^{-n} \sin (x/n)dx$
(b) $\lim\limits_{n \rightarrow \infty}$ $\int_0^1$ $(1+nx^{2})(1+x^2)^{-n}dx$
(c) $\lim\limits_{n \rightarrow \infty}$ $\int_0^\infty$ $n \sin (x/n) [x(1+x^2)]^{-1}dx$
Any help is greatly appreciated.
Update: I think I have successfully worked out arguments for each of (a) and (b), so I am less concerned about answers/strategies to those parts. However, (c) seems more tricky than the others, so if anyone visiting today sees how to handle (c) (in particular, a nice-enough dominating function!), let me know as it would be greatly appreciated.

REPLY [4 votes]: For part (c), you can use the following inequality (which is not hard to verify)
$$
|\sin(x)| \leq x\quad  \forall x\in[0,\infty)
$$
Then, we can get a dominating function as follows
$$
\left|\frac{n\sin(x/n)}{x(1+x^2)}\right| \leq \frac{n(x/n)}{x(1+x^2)} = \frac{1}{1+x^2}
$$