The existence of a limit as ε goes to zero and the independence of the final result from the regulator are nontrivial facts. The underlying reason for them lies in universality as shown by Kenneth Wilson and Leo Kadanoff and the existence of a second order phase transition. Sometimes, taking the limit as ε goes to zero is not possible. This is the case when we have a Landau pole and for nonrenormalizable couplings like the Fermi interaction. However, even for these two examples, if the regulator only gives reasonable results for ϵ ϵ ≫ ≫ ℏ ℏ c / Λ Λ (where Λ Λ is a superior energy cuttoff) and we are working with scales of the order of ℏ ℏ c / Λ Λ ′ , regulators with ℏ ℏ c / Λ Λ ≪ ≪ ϵ ϵ ≪ ≪ ℏ ℏ c / Λ Λ ′ still give pretty accurate approximations. The physical reason why we can't take the limit of ε going to zero is the existence of new physics below Λ.