When we try to regulate a divergent integral, we introduce another parameter, say $\lambda$ and then compute the integral. We finally take a limit (either $\lambda \rightarrow 0, \infty $) to restore the original situation. I have seen two outcomes of it.
On taking this limit, the divergences reappear. This happens in QED a lot. These divergences have then to be absorbed in the definition of physical (measurable) masses and charges in terms of the bare (unphysical) masses and charges.
On taking these limits, divergences do not appear and everything is fine. An example would be the Fourier transform of the Coulomb potential worked out here.
I want to know what goes on mathematically which makes these two situations different. Why do infinities reappear in the first case and not in the second? I am aware of Cauchy principal value integration if one wants to explain using this concept.