To answer your specific question, you should probably read either Knoebel's Exponentials Reiterated or McDonnell's Some Critical Points of the Hyperpower Function since they both talk about this.

Near zero, can be defined in one of three ways: as the limit of odd towers as the height tends to infinity (giving ), as the limit of even towers as the height tends to infinity (giving ), or as the inverse function of (giving ). Each one of these definitions gives a different answer as x approaches 0. The strange thing is that all definitions are equivalent for .

Andrew Robbins

Near zero, can be defined in one of three ways: as the limit of odd towers as the height tends to infinity (giving ), as the limit of even towers as the height tends to infinity (giving ), or as the inverse function of (giving ). Each one of these definitions gives a different answer as x approaches 0. The strange thing is that all definitions are equivalent for .

Andrew Robbins