|Oft discussed, oft erudite, oft esoteric, oft arcane: a paradox at the edge of infinity|
The edge of infinitesimal can be illustrated by the following comparison:
y=x ; y=0 ;;
y=2x ; y=0 .
Both systems, pairs of equations, have an intercept at [x y] = [0 0]. However, the first system instantly diverges while the second does not: For x=ð, infinitesimal not identically zero, the first diverges to [ð ð] on a slope of δ=1; while in the second [ð ²ð] the y-value is an infinitesimal of the x-value itself, on a slope of δ=0---challenging the notion that the two curves in the second system intercept at only one point:- when a slope of 0 is satisfied by an adjacent y-value actually =0, as the slope on either side is negative or positive and only zero can lie between and a lone point discontinuity would not have a slope. The finite interpretation is that a difference in x results in no difference in y for zero slope, and infinitesimal times zero should likewise not---not if 0.999... = 1.000.... But eventually y diverges infinitesimally when x exceeds infinitesimal, to the square-root of infinitesimal in this case,- though not yet finitesimal. Thus there is a tower structure in the transfinitesimal, qualification, range; and infinity is not the direct extension of arbitrarily large; and induction has a ways further to go: never crossing the chasm between finity and infinity, though mathematicians had long presumed.
Let's consider this more closely:
y = -1/x10 : x>0 ; y=0 ;
its first y equivalent to:
y = -n10 : n=1/x : x>0 ,
like the number 1.000...-0.999... at its n-th fraction-digit: this system converges
slower-still near x=0, And is discontinuous there: Its first y-derivative is,
-1/x10 / ²x£10 : x>0 , which is about
-n10..., but even as n→∞ and nears x=0 it never
reaches y=0 nor slope 0: for n to reach infinity, -its all,- is no sooner than 1/x and
its discontinuity ... Thus implicating a class principle of continuity, -versus a
distinction between 0.999... and 1.000... (as I have
preferred):- and a regradation among finitely vs. infinitely, Any, Each, Every, All.
A premise discovery under the title,