The limit in question is
$$\lim_{n\to \infty} \frac{c^n((2n-k-1)!)^2\frac{d^{k-1}sin(ln(x))}{dx^{k-1}}}{n!}$$
Where $c > 1$, $n$ and $k$ are integers and $1 \leq x \leq e^{\frac{\pi}{2}}$
What relation between k and n makes this limit less than or equal to 1? is there any k that satisfies this?