A Laplace transform can sometimes reveal the beginning or long-term value of a time-domain signal without computing the full inverse transform.

If

$$F(s)=\mathcal L{f(t)},$$

then the initial value theorem gives, under the usual regularity conditions,

$$f(0^+)=\lim_{s\to\infty}sF(s).$$

The final value theorem gives

$$\lim_{t\to\infty}f(t)=\lim_{s\to0}sF(s),$$

but only when the dynamics represented by $sF(s)$ are stable enough for a finite final value to exist.

The final value theorem must therefore not be used blindly. If the time-domain signal diverges or contains sustained oscillation, the limit may not exist even when the algebraic expression at $s=0$ appears finite.

These theorems connect endpoint behavior in time with limiting behavior in the Laplace domain and are especially useful for checking transient solutions and steady-state tracking.