How does Hawking radiation grow as a black hole evaporates?

The temperature of Hawking radiation is inversely proportional to the mass of a black hole, $T_{\rm H}\propto M_{\rm BH}^{-1}$, and so as the black hole shrinks the temperature of the radiation should grow. What happens as the mass shrinks to zero?

Naive application of the above formula leads to arbitrarily energetic Hawking radiation. Do assumptions in the Hawking Radiation derivation just break down in this limit? If so, is it known how to handle this limit?

-
Naive application of the above formula leads to arbitrarily energetic Hawking radiation. What do you mean by this? Any blackbody spectrum contains photons with arbitrarily high energies. –  Ben Crowell Oct 10 '13 at 1:04