Suppose $f\colon (0,1] \to \mathbb{R}$ admits derivatives of all orders on $(0,1)$ and is completely monotone: $$ (-1)^n f^{(n)}(x) \ge 0 \quad \text{for all $x\in (0,1)$ and $n \in \mathbb{N} \,\cup\, \{0\}$.}$$ Suppose further that $f(0) := \lim_{x \searrow 0} f(x) < \infty.$
I want to show that the limit of the first derivative $f'(0) := \lim_{x \searrow 0} f'(x)$ is finite; but so far, I could not think of a proof. I am sure that it would have to depend on the facts that $f(0) < \infty$ and that $f$ is decreasing.
Hint: Consider $f(x)=1-\sqrt{x}$.