I like most definitions. Maybe the ones that relate to notions of groupoid cardinality are a bit far out (nLab stuff).
I remember making a video about this
https://youtu.be/avc3Iv7Yojs>>11428287With all due respect, I think you're quite off. The limit definition was an early one, stemming from compound interest calculations. E.g. if you're raising an amount by 30% each week, for 5 weeks, then that's .
In fact, there's a cast to be made that the keeps residing at the core of the things. After the spectral theory- and before the anabelian Frenchy era, by the end of the war people had pretty much figured out most of the core topological groups things and the limit and forms of it pops up in various necessary conditions for when exponential function properties - such as
>>11428222 -- really still applies.
Translate limit of
to the equivalent one of
and now in denominator pops something up that relates to the resolvent of operations - such as the smooth propagation (also expressed in the differential equation form)
https://en.wikipedia.org/wiki/Resolvent_formalismAs far as intuition goes, the limit isn't that obscure from a functional analysis perspective. By the fundamental theorem of algebra, every non-constant polynomial naturally has zeroes to be found - while exponential function doesn't have zeros. How does it do it? Well one may start out with a naive approach constructing such as function. The polynomial (1-x) is struck by a zero at x=1. Well, then move it out. We can shift it out as far as we want: (1-x/n) has the zero at x=n. Well okay, but in the limit the function is just 1, the constant function. Thankfully however, (1-x/n)^n can be shown to remain finite and the limit is "a polynomial without zeros in a finite range"
