Infinite tetration and superroot of infinitesimal

[Ivars]

dx is used in the context of differentiation and integration. If you use it out of context, nobody knows what you mean. And if you additionally mix it into the theory of hyperreals, that doesnt simplify things. Limits however are well established in mathematics and
\( \lim_{x\downarrow 0} x^{1/x} = \lim_{x\downarrow 0} e^{\ln(x)/x} = 0 \) because \( \ln(x)/x \to -\infty \) for \( x\downarrow 0 \) (\( x\downarrow 0 \) means \( x\to 0 \) while \( x>0 \)).
There is nothing special about it.

In this thread, I was intereseted in infinitesimals in Leubniz/Euler sense, not limits. dx is an infinitesimal. What is dx^(1/dx)?

What is h(dx)? h(1/dx)? h(dx^2), h(1/dx^2)? etc.

If You use x>0, then You are of course right in evaluation of limit, but that is not considered as true limit, if values are different approaching from both sides, right?

What? The limit is a number and not a circle, how can a limit be a geometrical shape?

Undefined limits can be studied as limit cycles? To me this case (x^(1/x) looks similar to limit cycles in polar coordinates (even if we do not use complex values) , as even if it appraoches unit circle , it is not possible to tell where on Unit circle the end is as x-> infinity.
These spirals are perhaps similar to Gottfrieds lassoing with tetraseries, but I am not sure.

There has been suggestions to plot f(y,x) = y^(1/x) in 3D to see even better how it behaves close to y->0., x->0 . Or in Spherical. I just do not have the software/skills yet.

Ivars