Dilaton black holes with a pure electric charge are considered in a framework
of a grand canonical ensemble near the extreme state. It is shown that there
exists such a subset of boundary data that the Hawking temperature smoothly
goes to zero to an infinite value of a horizon radius but the horizon area and
entropy are finite and differ from zero. In string theory the existence of a
horizon in the extreme limit is due to the finiteness of a system only.Comment: 8 pages, RevTex 3.0. Presentation improved, discussion on metrics in
string theory simplified. To be published in Phys.Rev.
