The correspondence principle for mass-energy

Show that the equation E = mgc² obeys the correspondence principle.

As we accelerate an object from rest, its mass-energy becomes greater than its resting value. Classically, we interpret this excess mass-energy as the object's kinetic energy,

Expressing g as (1 - v²/c²)^-1/2 and making use of the approximation (1 + e)^p ≈ 1 + pe for small e, we have g ≈ 1 + v²/2c², so

which is the classical expression. As demanded by the correspondence principle, relativity agrees with classical physics at speeds that are small compared to the speed of light.