The famous Astrom-Hagander-Sternby theorem on limiting zeros of the pulse transfer function is extended by determining the accuracy of the asymptotic results for both the discretization and the intrinsic zeros when the sampling interval is small. Closed form formulas are derived that express the degree of the principal term of Taylor expansion of the difference between the true zeros and asymptotic ones as a function of the relative degree of the underlying continuous-time system and the value of the corresponding coefficient itself. Certain known results on asymptotic zeros are shown to be particular cases of the result presented.