# Initial Value Theorem of Laplace Transform/General Result

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## Theorem

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

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.

Let $\ds \lim_{t \mathop \to 0} \dfrac {\map f t} {\map g t} = 1$.

Then:

$\ds \lim_{s \mathop \to \infty} \dfrac {\map F s} {\map G s} = 1$

if those limits exist.