The super 0th root and a new rule of tetration?

[sheldonison]

Okay, I revised the formula, and I guess my intuation mislead me, maybe.
But I have a good news, too.
Maybe this formula will bring the trueth:
lim h->infinity (log(f(x+1/h)^^N)/log(f(x)^^N))^^h = lim h->infinity ((log(f(x+1/h))/log(f(x)))^^h)^^N
Do you think it can be correct?

I can't understand the formula, but one question is why have f(x) as opposed to x?  Are you trying to iterate f(x)???  

Also one assumes you are only interested in this limit for exp(-e)<=a<=exp(1/e), 
\( \lim_{h\to\infty}a\uparrow\uparrow h \) 

otherwise it is not defined.  if  exp(-e)<=a<=exp(1/e), then it is the real attracting fixed point of a^L=L.  Is this a correct understanding of your intentions?

update: Also, the 1/h terms in your equation drop out as h->infinity.  Then you have:
                (log(x^^N)/log(x^^N))^^h = ((log(x)/log(x))^^h)^^N

The ^^h is interpreted as the attracting fixed point.  The attracting fixed point of a^^infty as a approaches 1 also approaches a: 
fixed(a^^infity) ~=  a + (a-1)^2 + O(a-1)^3,
So the numerators equal the denominators and we are left with 1 = 1