Limits and the Index of a Square Matrix

For a square matrix A and a vector $b \ne 0$, over a topological Hausdorff field, necessary and sufficient conditions for the existence of the limits $\lim _{\varepsilon \to 0} \varepsilon ^m (A + \varepsilon I)^{ - 1} A^p $ and $\lim _{\varepsilon \to 0} \varepsilon ^m (A + \varepsilon I)^{ - 1} A^p \cdot A^p b$ are given in terms of the index of A, thereby extending some results of A. Ben-Israel [1]. Some characterizations of the class of square matrices which have index k are proven and the Drazin pseudoinverse of a square matrix is obtained as the limit $A^D = \lim _{\varepsilon \to 0} (A^{k + 1} + \varepsilon I)^{ - 1} A^k $, where k is greater than or equal to the index of A.