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Solve this limit
#1
Hi
solve this limit
I tried to sovle it but no way until now and I searched for solving formula but no results found.

this limit

[Image: gif.latex?lim_{x\rightarrow 0}(T(a,x))^\frac{1}{x}]

a>0

the result must be f(a)

if there is no soving formula,
then try to draw this function,
I don't know if there is a software support tetration!!

[Image: gif.latex?f(x)=(T(a,x))^\frac{1}{x}]

with various values of "a" and check the curve at x =0

thank you
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#2
(11/27/2012, 05:13 PM)Nasser Wrote: solve this limit
....

a>0 ...
What is T(a,x)? Perhaps super exponentiation base a of x? We usually say sexp(0)=1, which works for all bases. Also, in my quote, I modified your comment to use the tex tag.
- Sheldon

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#3
(11/27/2012, 06:28 PM)sheldonison Wrote: What is T(a,x)? Perhaps super exponentiation base a of x?
You are right
thank you
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#4
(11/27/2012, 05:13 PM)Nasser Wrote: solve this limit ....I don't know if there is a software support tetration!!
I posted a pari-gip routine that generates sexp(z) for real bases greater than here, http://math.eretrandre.org/tetrationforu...hp?tid=486.

By definition, T(a,0) = 1, since sexp(0) is defined to be 1. If T is analytic, then for each value of a, T has a Taylor series expansion around 0, corresponding to the Taylor series for sexp(z) around 0. Define as the first derivitive of that Taylor series.




There is an unproven conjecture that is analytic in the base=a for complex values of a, with a singularity at base . For real values of a, if , then sexp(z) goes to infinity at the real axis as z increases. If , then iterating converges towards the attracting fixed point as n goes to infinity, but this is a different function than tetration. Then for base>, we can have a taylor series for the any of the derivatives of , with the radius of convergence = .

I posted such a the taylor series for the first derivative of the base. For base=e, the first derivative ~= 1.0917673512583209918013845500272. The post includes pari-gp code to calculate sexp(z) for complex bases; the code for complex bases isn't as stable as the code for real bases, and doesn't always converge. If you're interested in a Taylor series for for your limit, search for "the Taylor series of the first derivative of sexp_b(z), developed around b=2" in this post: http://math.eretrandre.org/tetrationforu...e=threaded.
- Sheldon
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#5
You found an approximated solution.
It is ok, but this will not help me, because I tried to find the first derive of b^^x and x^^x and other related functions like for example b^^(x^2) by using differentiation fundamentals concepts, and I am just facing this problem to finish my work.
I may post my work here for discussion.

thank you Sheldonison.
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