We study the effect of die exit divergence on the sharkskin behavior, both experimentally as well as through the use of computational fluid dynamics (CFD) simulations. Sharkskin or surface fracture is known to occur immediately after the die exit, because of the large elongational deformation that a polymer melt experiences as it exits the die. We show that the diverging taper at the die exit postpones or completely removes the occurrence of sharkskin. The corresponding CFD simulations of an equivalent K-BKZ fluid show that the taper at the die exit significantly reduces the severity of the elongational flow, thereby reducing the normal stresses as the polymer melt leaves the die. We believe that, in an extrusion operation, the provision of a diverging taper at the die exit is one of the potential measures by which sharkskin instability can be eliminated on an industrial scale.