It's what you would consider a mathematical limit. The limit is never actually reached. V(t) becomes vanishingly small, to the point that time effectively stand

It's what you would consider a mathematical limit. The limit is never actually reached. V(t) becomes vanishingly small, to the point that time effectively stands still, but mathematically it should still always remain a non-zero value, and thus density at the singularity isn't infinite. Anyway, at that point measuring things becomes exceedingly difficult and abstract.