For a generic deformation of a two-dimensional holomorphic vector field with elementary degenerate singular point (saddle-node) we express the Martinet - Ramis orbital analytic classification invariants of the nonperturbed field in terms of the limit transitions between the linearizing charts of singularities of the perturbed field. In the case, when the multiplicity of the singular point of the nonperturbed field is equal to 2, we show that the Martinet - Ramis invariants are limits of transition functions that compare appropriately normalized canonic first integrals of the perturbed field in the linearizing charts. We prove a generalization of this statement for higher multiplicity singularities. For a generic deformation of higher-dimensional holomorphic vector field with saddle-node singularity we show that appropriate sectorial central manifolds of the nonperturbed field are limits of appropriate separatrices of singularities of the perturbed field. We prove the analogues of the two first results for Ecalle - Voronin analytic classification invariants of one-dimensional conformal maps tangent to identity.