First, a comment on "logical". Some of us are using it to mean "intuitive", some are using it to mean "accurate" and some of us are using it to mean "in accordace with the rules of logic".
First, the premise of relativity: "The speed of light in a vacuum is always a constant with respect to any given observer." This means exactly the same thing as "The speed of light in a vacuum is constant with respect to all observers." ALL of relative physics assumes this to be true without evidence of any kind whatsoever. It remains listed as a postulate.
The implications of a constant c are time/space distortion, absolute speed limits, and the like.
As for the higher math, let's review the properties of the extended real number system. It consistes of the real number system, plus either one or two points at infinity. For simplicity, I will focus on the system with two points, but limit myself to nonnegative numbers. I don't have access to a math typeset, so I will put everything in spoken form.
Brief overview of limits at infinity, infinite limits, and "continous at infinity": The limit at infinity is the same as the real number definition. If I say "the limit of f(x) as x approaches infinity is a" I mean "there is some b for which "if x>b, then |a-f(x)| < |c| for any nonzero c."" To say the limit is infinity means that there is some value of b for which if x>b, f(x)>c for all real c.
In briefer terms, I can get as close to the limit as I want, but not neccessarily reach the limit.
To say that a function is continous (on an interval) means that the limit of the function is equal to the value of the fuction on every point in that interval. That is to say, the point the function approaches is reachable at that point. A function that is undefined at a point is not continous on any interval containing that point.
Frankly, if you do not understand the behavior of limits better than I just explained it, you don't have the background to understand behavior at infinity.
Addition and multiplication not involving zero are continuous at infinity. That is to say, the limit as x approaces infinity of a+x and of a*x for x =/= 0 is equal to the value of x plus infinity or x times infinity. Most functions that are continous for all real numbers are continous at infinity. Divison, while continous at infinity, is NOT continous for all real numbers. Division by zero is undefined, and remains undefined. |x+2|(x+3)/|x+2| remains undefined for x=2 in the extended real number system. The form 0/0 remains inconclusive, and the form a/0 (for a=/= 0) remains undefined, but a/infinity and infinity/a ,(for a>0) are equal to 0 and inifinity, respectively.
Important note: This discussion pertains to math, only. No physical quantity that exists can ever be equal to infinity or zero. MOST physical quantities are not well-behaved approaching zero; that is the basis of quantam physics and molecular chemistry.