Are there an infinite amount of functions where the limit as z approaches infinity of f_prime^z(x) = 0? Are there an infinite amount of functions that you can take the derivative of an infinite amount of times without ever being equal to zero?
I'll try to clarify- if you take the derivative of f(x) an infinite amount of times and it doesn't equal zero, is there an infinite amount of functions f(x) could be? (A proof would be cool!)
f(x) = e^x is a possibility, as it is it's own derivative, and I believe the trigonometric functions work? Are there others?
Edit: Of different forms! Obviously the functions e^x, 2e^x, 3e^x... continues for infinity and all fit the description.

Do you consider the set of all polynomials to contain an infinite number of different forms? Because $e^xp(x)$ (where $p(x)$ is a non-zero polynomial) never vanishes no matter how many times you differentiate it. (You can easily prove this as the highest order term of $p$ stays the same.)
