Infinite tetration and superroot of infinitesimal

[bo198214]

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 something that is not properly defined, the answer will also not be properly defined. For my consideration the limit was the (properly defined mathematical) concept that comes closest to what you ask. Sometimes it seems to me that you intendedly try to avoid proper definitions because then the truth of your assertions could be verified.

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?

And infinitesimal (at least in the sense of hyperreals) is also positive or negative, isnt it? So you dont consider a both side limit when asking for dx^(1/dx).

Undefined limits can be studied as limit cycles?

I dont think so. Limit cycle is concept from dynamical systems and probably wrongly applied here.

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.

The limit \( x^{1/x} \) for \( x\to\infty \) is defined and so there is no need to complicate the matter.

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.

You dont get more information by putting it into spirals (at least not in the way you do it), everything that is needed to see for the limit, you can see already from the usual cartesian function graph.