Laplace Transform of Dirac Delta Function/Warning

Warning on Laplace Transform of Dirac Delta Function
Mathematically speaking, $\displaystyle \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} t$ does not actually exist.

Hence $\displaystyle \laptrans {\lim_{\epsilon \mathop \to 0} \map {F_\epsilon} t}$ is not actually defined.

However, it is useful to consider $\map \delta t = \displaystyle \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} t$ to be such that $\laptrans {\map \delta t} = 1$.