A concept of “near-minimax by characterization” is introduced, which formalizes the idea of an approximation “virtually indistinguishable from minimax.” Certain simple weighted approximation methods and related telescoping procedures, based on orthogonal polynomials, are then discussed in this context. In particular, it is found that, for a suitable $\beta ,e^{ - \lambda x} L_k^{ {- 1}/ 2} (2\lambda \beta x)$ is near-minimax by characterization in approximating the zero function on $[ {0,\infty } )$. It is hence easy to compute rational approximations, near-minimax by characterization, of a most appropriate form to $e^{ - x} $ on $[ {0,\infty } )$.