If $\mu$ is a positive, finite Borel measure on $\mathbb{R}$, and we have that for $a$ $\neq$ $0$ we have that $\mu (\mathbb{R})<\infty$
$f(a)= \int \frac{1}{\sqrt{x^{2}+a^{2}}}d\mu (x)$
Does $\lim_{a\rightarrow0 } af(a)$ exist?
I think this might be pretty simple, but I'm not entirely sure. I tried constructing a sequence of functions and used dominating convergence theorem to show that the limit is 0. Is that possible, or am I missing something?