According to the theory of the hierarchy of infinities, taking into consideration the first
quadrant of the Cartesian plane, and, assuming one draws the graph of the function
of an increasing oblique line
y = mx + q
of a parabolic function
y = ax ^ 2, with term a> 0
of a logarithmic function
y = log (x)
and of an exponential function
y = a ^ x
it is evident how the calculation of the limit for x which tends to + oo of all the
functions described above corresponds to + oo.
Always according to the theory of traditional mathematics, it is evident that the 4 functions mentioned above do not tend towards + infinity with the same speed, but with a displacement [delta (y)] different according to their intrinsic characteristics, although they tend, on the whole, to + infinity, without stabilizing at any point value of y, along their path.
It is always evident that the order of growth rate towards y, in the first quadrant, for x tending to + infinity, of the 4 functions is, from the fastest growing to the slowest growing, the following:
1st: exponential function
2nd: parabolic function
3rd: straight function
4th: logarithmic function.
Let us assume the result + infinity of the calculation of the limit for x that tends to + oo, that of an imaginary function that moves perfectly asymptotically towards the y axis. If the result of the calculation of the limit for x which tends to + infinity of the exponential function described above,
among the 4 functions considered is the one that comes closest to the characteristics of the pure solution + oo of the imaginary function described above, or in other words it is the one whose graphic path for x that tends to + oo moves more rapidly towards the y axis, then the result of the limit will be considered as + oo ^ (-). At this point, adding (-) at the apex of (+ oo) we
will have as results of the limits for x
which tends to + oo of the functions described above:
exponential function: + oo ^ (-)
parabolic function: + oo ^ (–)
linear function: + oo ^ (—)
logarithmic function: + oo ^ (—-).