Hey Paze.
If you allow discontinuities then the fastest growing function at a particular point is the delta function at x = 0.
Dirac delta function - Wikipedia, the free encyclopedia
Sure, it grows faster at first. But what happens for ? . Then what happens when we increase x by 1? for all x. In contrast, as because the numerator and the denominator are polynomials of the same degree with the same leading coefficient 1. For example, . Thus, when x is increased by 1, is always multiplied by , while is multiplied by smaller and smaller numbers that tend to 1.
This is clever, though delta function is not really a function. Even the piecewise function grows infinitely fast at 0.
If we restrict ourselves to continuous functions or to functions on natural numbers, suppose we have a candidate for the fastest-growing function . Then what about ?
I don't believe there is such a thing as "the fastest growing function," as you could always multiply whatever you think is the fastest function by 2 to get one that has a greater slope. However, in thinking about "normal" functions that grow very fast is a pretty good one. Once you get past about x=2.1 it grows much faster than . Of course the next logical extension of this is to consider , which grows so fast that it exceeds one googol around x= 3.84, then , etc, etc.