  We study the vanishing viscosity limit of the one-dimensional Burgers
equation near nondegenerate shock formation. We develop a matched asymptotic
expansion that describes small-viscosity solutions to arbitrary order up to the
moment the first shock forms. The inner part of this expansion has a novel
structure based on a fractional spacetime Taylor series for the inviscid
solution. We obtain sharp vanishing viscosity rates in a variety of norms,
including $L^\infty$. Comparable prior results break down in the vicinity of
shock formation. We partially fill this gap.
