Generalized recursive operators

[GFR]

So what are the exact values for :

\( \lim_{b\rightarrow-\infty}(e \begin{tabular}{|c|}\hline 7 \\\hline\end{tabular} b) = ? \)

\( \lim_{b\rightarrow-\infty}(e \begin{tabular}{|c|}\hline 9\\\hline\end{tabular} b) = ? \)

\( \lim_{b\rightarrow-\infty}(e \begin{tabular}{|c|}\hline 11\\\hline\end{tabular} b) = ? \)

etc?

Well, always for \( b < 0 \) let us start from (Hey, boys, I am starting using TeX ... Waaaoow!):

\( \lim_{b\rightarrow-\infty}(e \begin{tabular}{|c|}\hline 7 \\\hline\end{tabular} b) = c \)

For an estimation of \( c \), I think we may proceed like this, assuming the new more precise asymptotic value obtained by Andydude (step 0):

\( \lim_{b\rightarrow-\infty}(e \begin{tabular}{|c|}\hline 5 \\\hline\end{tabular} b) = -1.85.. \), and:

\( \lim_{b\rightarrow-1.85..}(e \begin{tabular}{|c|}\hline 5 \\\hline\end{tabular} b) = -\infty \)

Then:

step 1 - Calculate \( y = e \begin{tabular}{|c|}\hline 5 \\\hline\end{tabular} b \) vor various values of \( b < 0 \) and produce a graphical continuous and smooth plot, if possible;

step 2 - Produce the graphical inversion of the previous plot, always in the \( b < 0 \) domain, so obtaining the pentalog of \( b < 0 \) ;

step 3 - Estimate the intersection between the two pentation/pentalog diagrams (always in \( b < 0 \)), which should coincide with one the pentation fixpoints for \( y = b \). The coordinate of this point should be what we may call \( h \). This \( h \) is the coordinate of the penta-fixpoint and, therefore of the hexation asymptote. We can see thay it must be \( h < -1.85.. \).

step - Proceed again as in step 3 with the hexation/hexalog plots and the result will be \( c < h \), such that:

\( \lim_{b\rightarrow-\infty}(e \begin{tabular}{|c|}\hline 7 \\\hline\end{tabular} b) = c \), and:

\( \lim_{b\rightarrow c}(e \begin{tabular}{|c|}\hline 7 \\\hline\end{tabular} b) = -\infty \)

And ... so on !!!!
However, ... it is a very long way ... Perghaps there are shortcuts.

GFR
[Sorry, Administrator, these comments of mine, apart the first 5 lines are completely wrong. I shall correct them asap. Perhaps, I was tired! - GFR]