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+--- Thread: Limit when x approaches 0 (/showthread.php?tid=1774)
Limit when x approaches 0 - saudinho - 09/29/2023
Hello!
I browsed through the thread titles, but could not find anything related. I really apologize if this has already been asked before on this forum, but, here we go:
How can I rigorously prove that, for every natural n, the limit of x tetrated to n when x approaches 0 (through the positive side) is equal to 1 if n is even and 0 if n is odd?
I tried to prove through induction, but I never manage to conclude the proof. I also searched for it, but never found the proof of this
affirmation. It is stated on this article: https://math.osu.edu/sites/math.osu.edu/files/chun_tetration.pdf (page 5), and it says "evidences on the next section", but I still couldn't work it out.
Thank you.
RE: Limit when x approaches 0 - Daniel - 09/30/2023
(09/29/2023, 02:30 AM)saudinho Wrote: Hello!
I browsed through the thread titles, but could not find anything related. I really apologize if this has already been asked before on this forum, but, here we go:
How can I rigorously prove that, for every natural n, the limit of x tetrated to n when x approaches 0 (through the positive side) is equal to 1 if n is even and 0 if n is odd?
I tried to prove through induction, but I never manage to conclude the proof. I also searched for it, but never found the proof of this
affirmation. It is stated on this article: https://math.osu.edu/sites/math.osu.edu/files/chun_tetration.pdf (page 5), and it says "evidences on the next section", but I still couldn't work it out.
Thank you.
Well I suspect you are on the right track. The point 0 is inside a period two fractal is shown in the following red disk.
![[Image: FRACT010.gif]](https://tetration.org/original/Fractals/Atlas/Pseudocircle/FRACT010.gif)
RE: Limit when x approaches 0 - saudinho - 10/12/2023
(09/30/2023, 04:23 AM)Daniel Wrote:(09/29/2023, 02:30 AM)saudinho Wrote: Hello!
I browsed through the thread titles, but could not find anything related. I really apologize if this has already been asked before on this forum, but, here we go:
How can I rigorously prove that, for every natural n, the limit of x tetrated to n when x approaches 0 (through the positive side) is equal to 1 if n is even and 0 if n is odd?
I tried to prove through induction, but I never manage to conclude the proof. I also searched for it, but never found the proof of this
affirmation. It is stated on this article: https://math.osu.edu/sites/math.osu.edu/files/chun_tetration.pdf (page 5), and it says "evidences on the next section", but I still couldn't work it out.
Thank you.
Well I suspect you are on the right track. The point 0 is inside a period two fractal is shown in the following red disk.
Thank you for your answer. I tried to investigate these fractals, but I still couldn't solve it. What do you exactly mean the point is inside a two period fractal? And how does that relate with the limit? (I'm sorry if my question sounds too elemental).