Ok., let's do this in case of special relativity where we start with setting the speed of light c equal to 1. The starting point is then a setting where space and time have the same dimensions. As a consequence all speeds are dimensionless. We want to derive the classical limit from this setting.
Let's look at the expression for energy and momentum:
E = \gamma m
\vec{P}= \gamma m \vec{v}
where
\gamma = \frac{1}{\sqrt{1-v^2}}
Suppose we are interested at what happens at very low velocities, e.g. collisions between massive objects at extremely low velocities (note that v is dimensionless, so we can call velocities small in an absolute sense). We can then simply expand the above equations in powers of v and keep only the leading velocity dependent parts:
\vec{P}= m \vec{v}
E = m + 1/2 m v^2
These relations are not exact, we have ingored higher order tems in v. Now, the classical limit is not obtained by simply letting v tend to zero, as then the momentum becomes zero and the energy is equal to the mass and then nothing interesting is visible. Instead, when we approach the low velocity regime we need to zoom in by rescaling the velocity, so that when we approach the limit at which v has gone to zero, we still have a finite rescaled velocity. This means that phenomena that are infinitessimally small in the v = 0 limit remain visible.
So, let's introduce a rescaled velocity \vec{v}_{r} by putting:
\vec{v}= \frac{\vec{v}_{r}}{c}
where c is an arbitrary rescaling constant. We can now approach the limit of v to zero by keeping v_{r} fixed and let c go to infinity, so the velocity does not become invisible. In terms of \vec{v}_{r}, the equations become:
\vec{P}= m \frac{\vec{v}_{r}}{c}
E = m + 1/2 m \frac{v_{r}^2}{c^2}
We see that in the limit c to infinity the momentum becomes zero, so it makes sense to define a rescaled momentum:
\vec{P}_{r} = \vec{P} c
Then, in terms of rescaled variables, we have:
\vec{P}_{r}= m \vec{v}_{r}
We can then take the limit of c to infinity and still have a relation between finite quantities. In this limit, the higher order tems we have neglected are exactly zero.
If we look at the energy equation, we see that the energy becomes the mass in the limit c to infinity. Since energy is conserved, this means that the total mass is conserved in the classical limit (one thus assumes that when we let c go to infinity the rescaled velocities stay finite). The kinetic energy is given by:
E_{k} = E - m = 1/2 m \frac{v_{r}^2}{c^2}
This goes to zero in the limit c to infinity, so it makes sense to rescale the kinetic energy by multiplying it by c^2:
E_{k,r} = E_{k} c^2 = 1/2 m v_{r}^2
Then we can call the rescaled kinetic energy simply "the energy", and the original energy can be called the mass. Conservation of energy thus implies conservation of mass and of kinetic energy separately.
Note that in the original theory mass was simply the rest energy and not an independent physical quantity. But in the rescaled theory when we let the rescaling parameter tend to infinity mass has become independent of energy.
