  Using a unified approach we study the entropy of extremal black holes through
the entropy of an electrically charged thin shell. We encounter three cases in
which a shell can be taken to its own gravitational or horizon radius and
become an extremal spacetime. In case 1, we use a non extremal shell, calculate
all the thermodynamics quantities including the entropy, take it to the horizon
radius, and then take the extremal limit. In case 2, we take the extremal limit
and the horizon radius limit simultaneously, i.e., as the shell approaches its
horizon radius it also approaches extremality. In case 3, we build first an
extremal shell, and then take its horizon radius. We find that the
thermodynamic quantities in general have different expressions in the three
different cases. The entropy is the Bekenstein-Hawking entropy $ S=A_+/4$
(where $A_+$ is the horizon area) in cases 1 and 2, and in case 3 it can be any
well-behaved function of $A_+$. The contributions from the various
thermodynamic quantities for the entropy in all three cases are distinct.
Indeed, in cases 1 and 2 the limits agree in what concerns the entropy but they
disagree in the behavior of all other thermodynamic quantities. Cases 2 and 3
disagree in what concerns the entropy but agree in the behavior of the local
temperature and electric potential. Case 2 is in a sense intermediate between
cases 1 and 3. Our approach sheds light on the extremal black hole entropy
issue.
