Entropy of extremal black holes from entropy of quasiblack holes

The entropy of extremal black holes (BHs) is obtained using a continuity argument from extremal quasiblack holes (QBHs). It is shown that there exists a smooth limiting transition in which (i) the system boundary approaches the extremal Reissner-Nordström (RN) horizon, (ii) the temperature at infinity tends to zero and quantum backreaction remains bounded on the horizon, and (iii) the first law of thermodynamics is satisfied. The conclusion is that the entropy S of extremal QBHs and of extremal BHs can take any non-negative value, only in particular cases it coincides with S=A/4. The choice S=0 with non-zero temperature at infinity is rejected as physically unsatisfactory.