the final value theorem
The mirror image of the initial value theorem asks the opposite question: where does the function eventually settle? If a system reaches a steady state, the final value theorem reads that long-run value off the transform F(s) directly — by looking at F(s) as s approaches zero, the small-s end of the s-domain.
The theorem says: limit as t goes to infinity of f(t) equals limit as s goes to zero of s F(s). The pairing is the reverse of the initial value theorem: small s in the transform corresponds to large t in time, because the slowest-decaying part of e^(-st) (the part that survives longest) is governed by s near zero. So s F(s) evaluated near s = 0 captures the function's ultimate destination. You again multiply by s to undo the 1/s scale, but now you take the limit downward to zero instead of up to infinity.
This is enormously useful in control and circuits: it gives a system's final, settled output (the steady-state response) in a single step, without computing the full transient history — the steady error of a controller, the final charge on a capacitor, the settling temperature. But it carries a strict warning, because it silently produces a wrong number for any system that does not actually settle.
For X(s) = 5/(s(s + 2)) (a system settling to a constant), s X(s) = 5/(s + 2), and as s → 0 this → 5/2, so x(∞) = 2.5.
Multiply by s, send s to zero, and the steady-state value appears — provided the system really settles.
The final value theorem is only valid if s F(s) has all its poles strictly in the left half-plane (apart from a single allowed pole at s = 0). Applied blindly to an oscillating or growing system it returns a confident but meaningless number — for example, it would 'predict' that sin(t) tends to 0.