Infinite tetration and superroot of infinitesimal

[bo198214]

r= phi^(1/phi) , r= phi^(1/phi)^phi, r= phi^(1/phi)^phi^(1/phi) etc
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Now we think that infinity^(1/infinity) which was the beginning of this thread is not very well defined.
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So why not define the limits using the finite tangent angles? Or their trigonometric functions, e.g tangens? The one of the outgoing/incoming spiral for infinitesimal^(1/infinitesimal), the one of the limit circle at crossing for infinity^1/infinity.

There are options to define those limits , why no one is used?

I dont understand why you invoke a polar graph here, the limits are very well defined and well-known:
\( \lim_{x\to\infty} x^{1/x} = 1 \)
\( \lim_{x\to 0} x^{1/x} = 0 \)
You can already guess it when you look at the (cartesian) graph of the selfroot.

\( \lim_{x\to\infty} x^{(1/x)^x} = \lim_{x\to\infty} x^{1/x^x} = \lim_{x\to\infty} e^{\ln(x)/x^x}=e^{\lim_{x\to\infty}\ln(x)/x^x
}=e^0=1 \)
\( \lim_{x\to\infty} x^{(1/x)^{x^{1/x}}}=\exp\left(\ln(x)\frac{1
}{x^{x^{1/x}}}\right) \)

Here we know that \( x^{1/x}>1 \) for \( x>1 \) and hence \( x^{x^{1/x}}>x \) for \( x>1 \) and hence \( \lim_{x\to\infty}\ln(x)/x^{x^{1/x}}=0 \). Finally again:
\( \lim_{x\to\infty} x^{(1/x)^{x^{1/x}}}=1 \)