I think you are the one who does not understand infinity here, Clyde. Infinity can only be considered as existent in the context of topological space or when it used in measuring the "sizes" of sets, as I mentioned earlier. In a topological set, infinity is defined as the object which the set of real numbers converge to. You last statement (the one about how infinities can cancel out) is incorrect, because even in topological space, the infinite object cannot be operated upon in the same fashion as the real set. When someone writes '1/infinity=0' you are not really dividing one by infinity. Instead, it is a statement dealing with the series 1/1,1/2,1/3,1/4...and saying that it converges to 0. Your example 200/infinity is exactly the same.Your problem is you don't understand infinity; there IS a difference between 1/ infinity and 200 / infinity, they both give an infinitely small number, BUT those two numbers are different. Yes, you can have two infinite numbers and one can be larger than the other. This scenario arises quite often: consider integrating between 0 and infinity a y = x line, the area will be infinite, now consider integrating between 0 and infinity a y = 2x line, the area will also be infinite but a "larger" infinity than the first one.
Infinities can cancel out, infinitely large numbers multiplied by infinitely small numbers yield constants. (In some senses that is why light has a mass)
Your integration example is also faulty. To the extent that is meaningful to integrate any function with an infinite limit (improper integration), the result of integrating y=x from 0 to infinity is nonexistent, because it is divergent.