Is there a problem which can be solved in $O(f_k(n))$ time for any $k$ where $\lim_{k \to \infty} f_k(n) = 0$ but no constant time solution is known?
You can sort numbers in $O(\frac{n\log n}{k})$ for all fixed $k$, and $\lim_{k\to\infty} \frac{n\log n}{k} = 0$ for all fixed $n$.