Math overflow question on fractional exponential iterations

[sheldonison]

... Let a and b be real numbers above e^{1/e}, and c<d be real numbers.  Do we have 
\( \lim_{x \to \infty}\;\frac{\exp_a^c(x)} {\exp_b^d(x)} \;= 0 \)

Another observation is that if c=1, c<d then limit holds for all versions of tetration, even if a<b, assuming the derivative is strictly increasing.  If c=d=1, and a<b, then we would have 
\( \exp_a(x)<\exp_b(x) \)
but if \( d=1+\delta \) so d is a little bit bigger than c=1, then eventually for large enough values of x even though a<b.
\( \exp_a^{(1+\delta)}(x)\;>\;\exp_b(x)\;\;\; \) but only if x is big enough when a<b.
I think I could come up with an approximation for how big x has to be for this equation to hold.

This line of reasoning sheds some light on the general question which includes when c<>1, and helps one understand why Walker's solution works.