What is the final value theorem in Laplace domain and what condition must be satisfied?

• A. lim_{t→∞} x(t) = lim_{s→∞} s X(s); requires all poles of s X(s) lie in the left-half plane.
• B. lim_{t→∞} x(t) = lim_{s→0} s X(s); requires all poles of s X(s) lie in the right-half plane.
• C. lim_{t→∞} x(t) = lim_{s→0} X(s); requires ROC to include the imaginary axis.
• D. lim_{t→∞} x(t) = lim_{s→∞} X(s)/s; requires X(s) to be analytic everywhere.

Answer: (A)

The final value theorem connects the long-term value of a time-domain signal to its Laplace transform by looking at the s-domain limit of s X(s). Specifically, the steady-state value of x(t) as t grows without bound is given by the limit of s X(s) as s tends to infinity, and this holds only if the system is stable enough that all transient terms die out. The required condition is that all poles of s X(s) lie in the left-half of the complex plane, which ensures those transients are damped and do not prevent a finite limit. In this framing, the correct choice reflects both the s-domain expression and the stability requirement. The other options mix up the s-domain limit, omit the necessary extra factor of s, or state pole conditions that would allow non-decaying or divergent behavior, which is why they don’t match the final value theorem in this context.