The continuous-time persistent disturbance rejection problem (L(1) optimal control) leads to nonrational compensators, even for SISO systems [4], [7], [8]. As noted in [4], the difficulty of physically implementing these controllers suggest that the most significant applications of the continuous time L(1) theory is to furnish bounds for the achievable performance of the plant. Recently, two different rational approximations of the optimal L(1) controller were developed by Ohta ct al. [6] and by Blanchini and Sznaier [1]. In this paper we explore the connections between these two approximations. The main result of the paper shoes that both approximations belong to the same subset Ohm(T) Of the set of rational approximations, and that the method proposed in [1] gives the best approximation, in the sense of providing the tightest upper bound of the approximation error, among the elements of this subset. Additionally, we exploit the structure of the dual to the L(1) optimal control problem to obtain rational approximations with approximation error smaller than a prespecified bound.