Question

The final value theorem is used to find the

• A. steady state value of the system output
• B. initial value of the system output
• C. transient behavior of the system output
• D. none of these

Hint:

Final value theorem gives information about system behavior as time approaches infinity.

Answer 1

The Correct Option is A

This is a definitional question about what the final value theorem computes, so here is the reasoning behind ruling three options out and keeping one.
The theorem's formula, \( \displaystyle\lim_{t\to\infty} y(t) = \lim_{s\to 0} sY(s) \), explicitly evaluates the limit of the time response as \( t \to \infty \), which by definition is the steady-state (settled, long-term) value of the output.
The initial value, by contrast, comes from a separate theorem using \( s \to \infty \), and transient behaviour requires tracking the full time response rather than one endpoint value, so neither of those matches what this specific theorem computes. \[ \boxed{\lim_{t\to\infty} y(t) = \lim_{s\to 0} sY(s) = y(\infty)\ (\text{steady state})} \]

Answer 2

A complementary way to confirm this is to consider what physical question each of the four candidate purposes answers, then see which one matches the actual formula used by the theorem.

1. Option "steady state value of the system output": This describes the value \( y(t) \) settles to after a very long time, once transients have died out; the final value theorem's formula, evaluated at \( s \to 0 \) (corresponding to the zero-frequency, i.e. DC, or long-term behaviour of the signal), computes exactly this settled long-term value.
2. Option "initial value of the system output": This instead asks what the output equals right at the start of the response; that corresponds physically to the high-frequency (large \( s \)) content of the signal, which is why a separate theorem uses \( s \to \infty \) rather than \( s \to 0 \) for this purpose.
3. Option "transient behavior of the system output": This asks about the entire path the output takes while approaching its final value, which depends on all of the system's pole locations and their associated decaying exponential terms, not on a single evaluated limit; the final value theorem discards all of that information and keeps only the endpoint.
4. Option "none of these": Because the theorem's formula directly matches the "steady state" description in both its mathematical form and its physical meaning, no alternative description is needed.

Linking the theorem's mathematical limit (as \( s\to 0 \), i.e. long-term/DC behaviour) to its physical meaning identifies exactly one of the four candidate purposes.

So the correct answer is steady state value of the system output.