Initial Value Theorem of Laplace Transform/General Result

Theorem
Let $f$ and $g$ be real functions.

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$. Let $\displaystyle \lim_{t \mathop \to 0} \dfrac {\map f t} {\map g t} = 1$.

Then:
 * $\displaystyle \lim_{s \mathop \to \infty} \dfrac {\map F s} {\map G s} = 1$

if those limits exist.