# Tetration Questions

1. Dec 29, 2012

### pierce15

Hello all, of late I have been very interested in tetration (http://en.wikipedia.org/wiki/Tetration). I have a few questions that weren't really explained clearly by this wikipedia article. P.S., I can't get latex to work with the wikipedia notation, so I will use this notation: $x^{\Delta n}$.

1. Why does
$$\lim_{n\to\infty} x^{\Delta n}$$ only converge for $$1/e<x<e^{1/e}$$

2. How can the exact value be calculated as a limit? For example, how could I prove that:
$$\lim_{x\to\infty} \sqrt{2}^{\Delta x}=2$$

3. Why is the following statement true (from the wikipedia article):
$$\lim_{x\to0}\lim_{n\to\infty}x^{\Delta n}=1/e$$

I will be sure to post more questions as they come.

Last edited: Dec 29, 2012