The question is: Find the following limit
$$\lim\limits_{\epsilon \rightarrow 0} \int_0^{\infty} \frac{\sin{x}}{x}\arctan{(\frac{x}{\epsilon})}dx$$
I know that if I can move the limit in the integral the arctangent will become $\frac{\pi}{2}$ and then the remaining problem is just the Dirichlet integral. However, I am having trouble justifying that step.
Trouble justifying moving the limit inside the following integral
analysislimits
Best Answer
How about defining a sequence of functions $f_n(x)=\frac{sin(x)}{x} arctan(nx)$ where n is the integer part of $\frac{1}{\epsilon}$. As was suggested in the comment the real question is the interchange of the two limits, one for the improper integral and the other for $\epsilon$ (or n with the change of variables).
If you can show that $$\int_{0}^{b} f_n(x) dx$$ converges uniformly then we are done.
Try and show this with $\| \cdot \|_{\infty}$