Let $f:[1,\infty)\to\mathbb R$ be infinitely smooth and suppose $f$ and its derivatives decay faster than any power of $x$ as $x\to\infty$. In other words, $f\in\mathscr S([1,\infty)),$ where $\mathscr S$ denotes Schwartz space.
Then does $\lim_{a\to 0}\int_1^\infty f\left(\sqrt{a^2+x^2}\right)dx=\int_1^\infty f\left(x\right)dx$?
I would expect yes, and that a dominated convergence argument should apply somehow. I am tempted to use identities like $x<\sqrt{a^2+x^2}<a+x,$ except I can't assume $f$ is increasing/decreasing—it could even be oscillatory at infinity.
This is a "real" problem, by the way. I have been assuming this fact in my research, but am not sure how to prove it (or even if it is true).
Since $f$ is rapidly decaying $x^{4} f(x)$ is bounded. If $|f(x)| \leq \frac M {x^{4}}$ the $|f(\sqrt {a^{2}+x^{2}})|$ is bounded by $\frac M {x^{2}}$ which is integrable in $(1,\infty)$.