What is the speed of light in geometrical optics? Geometrical optics or Hamiltonian optics is the short wave length limit of Maxwells equations of light. In Hamiltonian form it is equivalent to the Hamiltonian description of a single, classical, non-relativistic particle with mass $m$. 
However in non-relativistic, classical mechanics it is sometimes said, that the speed of light is assumed to be infinite. But I believe it must be assumed, that the speed of light is at least finite (but probably not invariant for inertial observers?) for the theory to make sense. But I'm not sure. What is the assumed speed of light within the framework of this theory?
 A: In the construction of geometrical optics, no assumption is made on the speed of light. One can construct the Fermat principle and associated approximations even in fully relativistic formalism. (An example can be found in the book Gravitation by Misner, Thorne and Wheeler.) 
On the other hand, if deriving non-relativistic mechanics from relativity, we do assume that the constituents of the mechanical system move at speeds $v$ much smaller than the speed of light $c$ in our lab frame. I.e. $v/c \approx 0$ and our lab frame plays a privileged role.
The fact that both classical mechanics and ray optics both have a Lagrangian and Hamiltonian formulation does not mean they are both ``nonrelativistic" in some sense or that they can be derived using some unified limit - they are not. In fact, even fully relativistic particle mechanics have a Lagrangian and Hamiltonian formulation. This is because the Lagrangian and Hamiltonian formalism is more of a mathematical method how to describe dynamics of a broad class of systems. 
But! Your are correct that using Hamiltonian optics and generally any light dynamics along with non-relativistic mechanics leads to a weird inelegant system. Namely, weird extra terms pop up e.g. in dispersion relations of light when transforming between reference frames using Galilean transformations. 
This was often historically hand-waived away by saying that the "nice" equations for the propagation of light are defined with respect to the reference frame of the medium and this also lead to the postulation of aether, the ``vacuum medium". Ultimately, this weird aether business and the Michelson-Morley experiment lead Einstein to his special relativity. 
So the "classical" or "non-relativistic" sets of theories for light and massive particles are not quite consistent and one cannot really expect them to be. (They are quite useful nonetheless...)
A: Similar to what Sammy Gerbil said, the speed of light doesn't have to be assumed, because the value of speed is c=3.00*10^8 m/s. However, it is disputed that different kinds of mechanics can cause different speeds of light. Like the chart below:

And similar to valerio92, Maxwell did realize that light and magnetism were connect together.  The speed of propagation in a magnetic field is the same as the speed of light.
hope this helps :)
