# How to find a function which satisfies such functional equation?

How to find a function which satisfies:

$$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$

where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 it's inverse function)?

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 Try x=0 to see if there is a typo. – T.. Nov 6 '10 at 1:52 So $f_a(x)=1 \text {for} x \neq 0$, undefined for $x=0$ works. – Ross Millikan Nov 6 '10 at 4:01 Your equation is unclear, because the exponent $h$ is written on the exterior of a number, not a function. – Phira Nov 25 '11 at 11:27