transfinitesimal qualification

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=, infinitesimal not identically zero, the first diverges to [ ] on a slope of δ=1; while in the second [ ²] 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,

Grand-Admiral Petry
'Majestic Service in a Solar System'
Nuclear Emergency Management

2003, 2006 GrandAdmiralPetry@Lanthus.net