tetration limit ??

[BenStandeven]

Let's see here.

The fixed point for base (eta + eps) is e + delta(eps), where delta satisfies:

\( \delta(\eps) = -e^2 \eps/\eta + \sqrt{2/\eta} e^{3/2} \sqrt{-\eps} + O(\eps^{3/2}) \)

Now \( (\eta + \eps)^{e + \Re(\delta(\eps))} = (\eta + \eps)^{e + -e^2 \eps/\eta + O(\eps^{3/2})} = \e( \ln(\eta + \eps) (e + -e^2 \eps/\eta + O(\eps^{3/2}))) \), which is:

\( \e( 1 + -e \eps/\eta + \ln(1 + \eps/\eta) (e + -e^2 \eps/\eta) + O(\eps^{3/2})) = \e( 1 + O(\eps^{3/2})) = e + O(\eps^{3/2}) \).

So:

\( (\eta + \eps)^{e + \Re(\delta(\eps)) + \theta} = (\eta + \eps)^{(e + \Re(\delta(\eps)))(1 + \theta/(e + \Re(\delta(\eps))))} = e^{(1 + O(\eps^{3/2}))(1 + \theta (1 - \Re(\delta(\eps))/e + O(\eps^2))/ e )} \) which for \( |\theta| << 1/\sqrt{\eps} \) is \( e^{1 + \theta (1 + e \eps/\eta)/ e + O(\eps^{3/2}) } = e^{1 + \theta (1 + e \eps/\eta)/e} + O(\eps^{3/2}) \)

To be continued...