Limit when x approaches 0

[saudinho]

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/...ration.pdf (page 5), and it says "evidences on the next section", but I still couldn't work it out.

Thank you.

[Daniel]

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/...ration.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.

[saudinho]

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/...ration.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).

[leon]

The colors are for signed area as it approaches and doesn't approach white as much because of how much color there is

[leon]

The limit is as many x as there are regions like numbers like complex numbers and the boundary is the graph which has an x