For some function $f$ and $g$ lets say that I have an integral which looks like, $\int ^{f(\epsilon)}_0 g(t,\epsilon) dt$. So if I want to compute this to zeroth order in $\epsilon$ can I just truncate the Taylor series of $f$ and $g$ to the zeroth order in $\epsilon$ and get the answer?
(..so as a special case if the Taylor series of $f$ has no term in the zeroth order of $\epsilon$ then shouldn't the answer be $0$ no matter what is $g$?..)
I am asking this because $g$ is so complicated that the full integral can't be done. So I don't have the luxury of doing the full integral and then taking the $\epsilon \rightarrow 0$ limit.