Ivars Wrote:Question: What about other superroots of 1? 3rd, 4th, n-th, x-th, zth etc? If:
The answer to this is really easy, you just need to look at the graph.
It is well known that \( \lim_{x\to 0} {^n x} \) is 0 for odd \( n \) and 1 for even \( n \).
It follows that the solution of \( {^n x}=1 \) is only 1 for odd \( n \) and \( \{0,1\} \) for even \( n \).
Quote:\( x^{(x^x)} = 1 \) then as \( 0^0=1 \), \( x^0=1 \) in this
case 3rd superrot of y=1 is also x=0
\( 0^{0^0}=0^1=0 \).
(though note that statements like \( 0^0=1 \) can be quite wrong, for example when considering \( \left(e^{-x^2}\right)^{1/x} \). If you naively set \( \lim_{x\to\infty}=0^0=1 \) this is wrong because \( \lim_{x\to \infty} e^{-x^2 * x} = \lim_{x\to\infty} e^{-x} = 0 \).)