In sparse linear regression, the SLOPE estimator generalizes LASSO by penalizing different coordinates of the estimate according to their magnitudes. In this paper, we present a precise performance characterization of SLOPE in the asymptotic regime where the number of unknown parameters grows in proportion to the number of observations. Our asymptotic characterization enables us to derive the fundamental limits of SLOPE in both estimation and variable selection settings. We also provide a computational feasible way to optimally design the regularizing sequences such that the fundamental limits are reached. In both settings, we show that the optimal design problem can be formulated as certain infinite-dimensional convex optimization problems, which have efficient and accurate finite-dimensional approximations. Numerical simulations verify all our asymptotic predictions. They demonstrate the superiority of our optimal regularizing sequences over other designs used in the existing literature.