02/06/2008, 02:44 PM
I think that the ordinates of the x -> -oo asymptotes of y = b[5]x are depending on the values of base b.
In particular, it can be shown that, for b = Eta = e^(1/e) = 1.444667861.., we have two fixpoints in y = b#x and that they probably (...) are:
x = {-Pi/2, +Pi/2} = {-1.570796327..., +1.570796327..}, giving two symmetrical horizontal asymptotes to the base-Eta pentation, y = Eta[5]x, with the same (positive, for x -> +oo, and negative, for x -> -oo) values.
So, infinite pentations may also be finite, but I presume that, for b slightly greater than Eta, the pentation will easily ... explode!
GFR
In particular, it can be shown that, for b = Eta = e^(1/e) = 1.444667861.., we have two fixpoints in y = b#x and that they probably (...) are:
x = {-Pi/2, +Pi/2} = {-1.570796327..., +1.570796327..}, giving two symmetrical horizontal asymptotes to the base-Eta pentation, y = Eta[5]x, with the same (positive, for x -> +oo, and negative, for x -> -oo) values.
So, infinite pentations may also be finite, but I presume that, for b slightly greater than Eta, the pentation will easily ... explode!
GFR