Actually, the correct formula is
or, solved for $E$,
If you Taylor expand $\gamma$ in powers of $v^2/c^2$, you will get
You see that in the limit of large $c$ only the first two terms survive since $v^2/c^2$ becomes very small. The second term is the well known kinetic energy. The first term is a fixed constant which is actually indeed infinite. However, you might have already heard that energy, as well as potential, is only defined up to an additive constant. Therefore, it is possible to shift the energy scale such that the constant first term vanishes:
This is what we have in non-relativistic mechanics.