  We will investigate the intersection of the normal operators with the closure
of the nilpotent and quasinilpotent operators in various C*-algebras. A
complete characterization will be given for type I and type III von Neumann
algebras with separable predual and for unital, simple, purely infinite
C*-algebras. Some restrictions that prevent normal operators from being limits
of quasinilpotent operators will be given. The case of AF C*-algebras will be
of particular interest. In addition, the closure of the span of the nilpotent
operators and the distance from a projection to the nilpotent operators of
various C*-algebras will be examined.
