I have only a rough idea about what you suggest, so please let me ask some more questions

Let , when we consider we realize that , and . So the critical region is at infinity, where there .

Vice versa and .

I still dont see how you will use this for defining .

What do you mean by "there are fewer and fewer degrees of freedom for how to extend the sexp function to the real numbers"?

And otherwise of course you surely read already Jay's post about change of base, which attributes the same effect you describe here that converges/should converge to a constant for if . Moreover he gives a formula to derive from any given such that exactly this condition is satisfied.

Let , when we consider we realize that , and . So the critical region is at infinity, where there .

Vice versa and .

I still dont see how you will use this for defining .

sheldonison Wrote:Consider what happens as b approaches e^(1/e) in the equation . The curve becomes more and more linear, and there are fewer and fewer degrees of freedom for how to extend the sexp function to real numbers, and still have an increasing "well behaved" function. It must be possible to describe this rigorously in terms of limits. Here is an example.

if , , , , , , ,

What do you mean by "there are fewer and fewer degrees of freedom for how to extend the sexp function to the real numbers"?

And otherwise of course you surely read already Jay's post about change of base, which attributes the same effect you describe here that converges/should converge to a constant for if . Moreover he gives a formula to derive from any given such that exactly this condition is satisfied.