Why can't the constancy of the speed of light be deduced from classical physics? I have read over a dozen questions about the speed of light -- "why it $c$ constant?", "why can't anything travel faster than light?", "how do we know this?"
The responses are quite clear:


*

*The invariance of light speed is determined empirically (e.g. from the Michelson-Morley experiment).

*The speed of light is simply an axiom for physics and was discovered experimentally.

*The invariant value of $c$ is a fundamental property of the universe.
My question is why can't the invariance of $c$ not be deduced theoretically with the following logic.


*

*As an object's velocity increases, its kinetic energy also increases.

*The kinetic energy growth is asymptotic, meaning it approaches infinity as the velocity approaches some value.

*This makes it impossible for anything with a mass to reach this velocity because it would require infinite kinetic energy.

*Therefore velocity must be have a limit.
See, this makes much more sense to me than the claim that the invariance of $c$ is just a postulate from lab work and that there's no reason for it to be invariant other than "it's just the way things are".
I suspect I've made a mistake. Perhaps the idea that an object's mass increases rapidly as speed increases comes from special relativity itself, which is derived from the assumption that $c$ is invariant. This doesn't, however, seem obvious to me since we should be able to observe this effect in experiments. 
 A: In classical mechanics, particles can have arbitrarily high speeds and still have finite energies, so your statement 2 fails. Given any finite speed $v$, the energy is $1/2 mv^2$. We can keep on increasing $v$ and the energy will increase and still remain finite.
Suppose you think the maximum speed is $C$. Then a particle with such a speed has energy $1/2mC^2$ (which is finite). Why can't a particle have speed $C + \delta C$, which has energy $1/2 m(C+ \delta C)^2$, which is still finite?
A: As you have phrased it, your argument appears to be circular, because the asymptotic approach of energy to infinity at some finite maximum velocity is is deduced from the existence of a finite maximum velocity. However, we can tighten up the reasoning.
Either there is, or there is not, a finite maximum speed in nature. If there is not, then we would have Newtonian relativity which is empirically false. Consequently there is a finite maximum velocity and special relativity follows. (Newton himself argued that instantaneous action at a distance was an absurdity. I am inclined to agree, but I don't use that as a basis for a logical argument). 
Note that relativity does not depend specifically on the speed of light, but only on the existence of a maximum speed. It just happens that, to the accuracy of measurement, light travels at the maximum speed.
The constancy of the maximum speed follows because, in natural units, all speeds are fractions of the maximum, and because, to carry out science at all, we must assume that the fundamental behaviour of matter is always and everywhere the same (Hume's principle of uniformity in nature).
A: As other posters have noted, your assumptions about classical mechanics are not correct.  Newtonian mechanics cannot predict a theoretical maximum speed.
What you are looking for in your argument is to solve Maxwell's equations for the speed of an electromagnetic wave, and find that the result is completely invariant at $$c=\frac{1}{\sqrt {\mu_0 \epsilon_0} }$$  This leads to either some sort of universal medium (aether?) with which c is constant relative to, or the start of the ideas in special relativity that c is constant in every frame, which then leads to the Lorentz equations.
A: 
The kinetic energy growth is asymptotic, meaning it approaches infinity as the velocity approaches some value.

Unfortunately, this result already assumes that you know that there is an invariant speed. Without the invariant speed the formula for KE is $KE=\frac{1}{2}mv^2$ which has no limiting speed. It is only after you already know about the invariant speed that you get the expression $KE=((1-v^2/c^2)^{-1/2}-1)mc^2$ which goes to infinity as $v$ approaches $c$. 

photons don't have mass so they can

This also requires already knowing about the invariant speed. Without the invariant speed there is no known relationship between all three of mass, energy, and momentum. With the invariant speed we learn $m^2 c^2=E^2/c^2-p^2$ which given the energy and momentum of light implies that light is massless. 
So yes, those things, if known independently somehow, would have led to the conclusion of an invariant speed. But how could they have been known? Perhaps they could have simply been measured and known experimentally first, but historically it didn’t happen that way. Historically the invariance of c was postulated prior to measurements experimentally showing those points. Furthermore, such measurements would have been considered violations of classical physics. 
A: In classical physics the kinetic energy of an object is
$$E_{\text{kin}}=\frac{1}{2}mv^2,$$
which clearly has no asymptotes. That is, the second step in you argument is wrong.
In special relativity the kinetic energy is 
$$E_{\text{kin}}= \left(\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}-1\right)mc^2,$$
which does have an asymptote. However, this formula is derived with the constancy of $c$ as a starting assumption.
