Charge to mass ratio of an electron
The electric charge on an electron is fixed, regardless of its speed. The apparent mass of an electron INCREASES with its speed (relative to the observer). So charge/mass will decrease as mass → ∞. The major problem with these statements is the use of "mass" to mean "apparent mass" or "observed mass". More and more (and probably the majority now days) of the Physics community reserve the term "mass" to mean "rest mass" (which is the mass observed at zero velocity relative to the observer). This is a bit more complicated than it seems: consider a gas. The center-of-mass velocity may be zero (relative) but obviously each molecule has its own velocity not dependent on center-of-mass. Fortunately, in the range of temperatures chemists are concerned with, relativistic velocities aren't seen, so no correction is necessary between a molecule's rest mass and its apparent mass. Apparent mass, observed mass, relativistic mass, dynamic mass, are (usually) all different ways of expressing the fact that the energy of an object affects its mass; E= mc² and that energy is the sum of potential and kinetic terms, where the kinetic term is ½mv². Hence the mass you will observe depends on how fast the object is traveling relative to you (or your observational equipment). The Special Relativity equation for mass is m = m0 ÷ √( 1 - (v/c)²) where v is the particle's speed (magnitude of its velocity), c is the speed of light, and m0 is its mass when v=0, i.e. its rest mass. As you can see, as v approaches c, the denominator approaches 0 and so the observed mass gets larger without limit.

I should also note that the generalization I made, that chemists need not "worry" about relativistic velocities, isn't entirely correct. For instance, the color of gold is due to relativistic effects (otherwise it would be silver just like the other more common metals). In general, atoms do not have relativistic velocities, but larger atoms have electrons with relativistic momentum, which can be thought of as relativistic speed. There's no reason why atoms can't have relativistic speeds, other than the obvious problem of how we'd contain and observe them. If there wasn't any "chemistry" at relativistic speeds, we'd have fusion power already - confinement of plasma is "hard" because it interacts with the container.