Laplace Transform of Dirac Delta Function/Lemma

Lemma for Laplace Transform of Dirac Delta Function
Let $F_\epsilon: \R \to \R$ be the real function defined as:


 * $\map {F_\epsilon} t = \begin{cases}

0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le t \le \epsilon \\ 0 & : t > \epsilon \end{cases}$

Then:
 * $\laptrans {\map {F_\epsilon} t} = \dfrac {1 - e^{-s \epsilon} } {\epsilon s}$