> Technically, 0^0 is an indeterminate form and has no specific solution.
Precisely, as this is the true mathematician answer: "it depends where 0^0 comes from".
As a f(x,y): RxR->R function, come from the top of the R² plane and 0^0 is 0 but come from the right side and it's 1. Limits and extension by continuity give us this easily enough for fh:x->x^0 and fv:y->0^y.
Writing this I asked myself, what if we came from some funky other path, like the diagonal, or a curve?
h: R->RxR, x->(x, 0) defines "coming from the top", and foh = fh
v: R->RxR, x->(0, y) defines "coming from the top", and fov = fv
d: R->RxR, x->(x, x) defines coming along the diagonal, where things could get interesting.
s: R->RxR, t->(e^(at)sin(t), e^(at)cos(t)) defines coming along a log spiral whose tangent at t=0 is vertical, so fos looks like fun around t=0.
Now what happens if we build a path function p: RxR->RxR, (t, z)->? that endlessly approaches v when z->0? the log spiral with z=1/a as a parameter is a possible one. With such a p function, what does lim fop(x) when x->0 (which is a function of z) look like when subsequently z->0?
Precisely, as this is the true mathematician answer: "it depends where 0^0 comes from".
As a f(x,y): RxR->R function, come from the top of the R² plane and 0^0 is 0 but come from the right side and it's 1. Limits and extension by continuity give us this easily enough for fh:x->x^0 and fv:y->0^y.
Writing this I asked myself, what if we came from some funky other path, like the diagonal, or a curve?
h: R->RxR, x->(x, 0) defines "coming from the top", and foh = fh
v: R->RxR, x->(0, y) defines "coming from the top", and fov = fv
d: R->RxR, x->(x, x) defines coming along the diagonal, where things could get interesting.
s: R->RxR, t->(e^(at)sin(t), e^(at)cos(t)) defines coming along a log spiral whose tangent at t=0 is vertical, so fos looks like fun around t=0.
Now what happens if we build a path function p: RxR->RxR, (t, z)->? that endlessly approaches v when z->0? the log spiral with z=1/a as a parameter is a possible one. With such a p function, what does lim fop(x) when x->0 (which is a function of z) look like when subsequently z->0?
Damn. It was supposed to be a two-line comment.