In this paper, we study the limiting properties of the K energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian ( W. Ding and G. Tian. KählerEinstein metrics and the generalized Futaki invariant. Invent Math, 110:315–335, 1992.) in the case of hypersurfaces. In particular, we allow the center fiber of a special degeneration (a degeneration by a ...