"If $f$ is integrable on $[0,A]$ for every $A>0$, and $\lim_{x\to\infty}f(x)=1$, then $\lim_{t\to 0^+} t\int_{0}^{\infty} e^{-tx}f(x)\: dx$ exists".
(I'm convinced it's true, some examples suggest that the answer is 1 for every function with these properties, but how to prove it?)