  The Bergman-Milton bounds provide limits on the effective permittivity of a
composite material comprising two isotropic dielectric materials. These provide
tight bounds for composites arising from many conventional materials. We
reconsider the Bergman-Milton bounds in light of the recent emergence of
metamaterials, in which unconventional parameter ranges for relative
permittivities are encountered. Specifically, it is demonstrated that: (a) for
nondissipative materials the bounds may be unlimited if the constituent
materials have relative permittivities of opposite signs; (b) for weakly
dissipative materials characterized by relative permittivities with real parts
of opposite signs, the bounds may be exceedingly large.
