  In previous work the authors considered the asymmetric simple exclusion
process on the integer lattice in the case of step initial condition, particles
beginning at the positive integers. There it was shown that the probability
distribution for the position of an individual particle is given by an integral
whose integrand involves a Fredholm determinant. Here we use this formula to
obtain three asymptotic results for the positions of these particles. In one an
apparently new distribution function arises and in another the distribuion
function F_2 arises. The latter extends a result of Johansson on TASEP to ASEP.
