I recently came across a formula that involved the assumption that $\lim_{n\to \infty}|f^{(n)}(x)|=0$, and at first I thought that it would be true for every function, but then I remembered that trigonometric functions like $\sin(x)$ have cyclical derivatives where after a certain number of times of differentiating $\sin$ you eventually get back to $\sin$ and that there are other functions like ${1 \over x}$ where $\lim_{n\to \infty}|f^{(n)}(x)|=\infty$. I started to think about the situation as $\lim_{n\to\infty}{d^nf \over dx^n}=0$ where $f$ can be found by repeated integration and ends up being equal to $\lim_{n\to\infty}\sum_{k=0}^nC_kx^k$ which I understand to be any polynomial. But function like $e^x$ have polynomial/series expansions with infinite radii of convergence but $\lim_{n\to \infty}|f^{(n)}(x)|\neq0$. I guess I have to restrict the category of to all finite polynomials for the limit to be true. But another interesting note is that for $a\in(1,e)$ if $f(x)=a^x$ then $\lim_{n\to \infty}|f^{(n)}(x)|=0$ is true, which tells me that it applies to some non-polynomials or alternately to some infinite polynomials.
Even though I've been able find a couple categories of functions were $\lim_{n\to \infty}|f^{(n)}(x)|=0$ is and isn't true I can't seem to find any special classification or underlying similarity for those that it does apply to, so I'm wondering if there is anything that can be said about functions were the limit equals zero in general or if the limit is the only thing that connects them.
As an end note I did a little research and came with one thing that they all have in common, they can be infinitely differentiated, surprise surprise!, which can also be stated as they are all smooth. I couldn't find any subcategorization of smooth functions relevant to my question.