Correct Answer - Option 3 : cannot be determined
Concept:
1. Final value theorem:
A final value theorem allows the time domain behavior to be directly calculated by taking a limit of a frequency domain expression
The final value theorem states that the final value of a system can be calculated by
\(f\left( \infty \right) = \mathop {\lim }\limits_{s \to 0} sF\left( s \right)\)
Where F(s) is the Laplace transform of the function.
For the final value theorem to be applicable system should be stable in a steady-state and for that real part of the poles should lie on the left side of s plane.
2. Initial value theorem:
\(C\left( 0 \right) = \mathop {\lim }\limits_{t \to 0} c\left( t \right) = \mathop {\lim }\limits_{s \to \infty } sC\left( s \right)\)
It is applicable only when the number of poles of C(s) is more than the number of zeros of C(s).
Calculation:
Given that, \(X(s) = \frac {12(s + 2)}{\{s(s^2 + 1)\})}\)
Poles lies at s = 0, ±j 1
Roots lies on the imaginary axis, so it is marginally stable.
So the Final value theorem is not applicable as the system is oscillatory in nature.
∴ The correct answer is option C 'cannot be determined'.
