  The monotonicity-based approach has become one of the fundamental methods for
reconstructing inclusions in the inverse problem of electrical impedance
tomography. Thus far the method has not been proven to be able to handle
extreme inclusions that correspond to some parts of the studied domain becoming
either perfectly conducting or perfectly insulating. The main obstacle has
arguably been establishing suitable monotonicity principles for the
corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this
shortcoming by first giving a convergence result in the operator norm for the
Neumann-to-Dirichlet map when the conductivity coefficient decays to zero
and/or grows to infinity in some given parts of the domain. This allows passing
the necessary monotonicity principles to the limiting case. Subsequently, we
show how the monotonicity method generalizes to the definite case of
reconstructing either perfectly conducting or perfectly insulating inclusions,
as well as to the indefinite case where the perturbed conductivity can take any
values between, and including, zero and infinity.
