TITLE: Is there a differential limit?
QUESTION [9 upvotes]: I'm wondering if there's such a concept as a "differential limit". Let me give an example because my nomenclature is my own and unofficial, but hopefully indicative of the concept.
For some function f(x), there may exist a derivative of that function f'(x) which we call a first order differentiation of f(x). Likewise from f'(x) we could also derive a second order function f''(x). Is there a mathematical concept of the limit of a function as the order of differentiation goes to infinity? 
I can think of a function that I can take the derivative of an infinite number of times ( albeit without getting a constant value): f(x) = sin(x)
The function g(x) = x^2 might have a differential limit of 0. (g'(x) = 2x; g''(x) = 2; g'''(x) = 0 ...)
Is there any usefulness to this concept? A topic of study or keyword that discusses this? Any more examples of infinitely derivable functions?
Most importantly if this is interesting to anyone other than myself: what is a good resource for learning more?

REPLY [6 votes]: It would seem that if the sequence $f$, $f'$, $f''$, ..., $f^{(n)}$, ... tends to a limit as a function, then that limit would be a solution of the differential equation $\frac{dy}{dx}=y$. So if the limit exists, it is either zero or some multiple of $e^x$. (I'm assuming that the notion of limit we're using is one that the differentiation operator is continuous with respect to, which doesn't seem like much to ask. Pointwise convergence is out, though).
Also, if we have some function whose iterated derivatives converge towards $ce^x$, we can subtract $ce^x$ from the initial function and get a new sequence whose iterated derivatives converge towards the zero function. So if we look for solutions to $\lim_{n\to\infty} f^{(n)}=0$ we essentially get all functions whose iterated derivatives have a limit.
All polynomials qualify, of course, but there are also non-polynomial analytic solutions such as $\sum_{n=0}^\infty \frac{1}{n\cdot n!} x^n$ and in general $\sum_{n=0}^\infty \frac{a_n}{n!}x^n$ whenever $\lim_{n\to \infty} a_n = 0$. I wonder whether some of these have nice closed forms.