# Final Value Theorem of Laplace Transform

Published on 24/2/2012 & updated on Saturday 14th of July 2018 at 06:02:38 PMFor the sake of example If F(s) is given, we would like to know what is F(∞), Without knowing the function f(t), which is Inverse Laplace Transformation, at time t→ ∞. This can be done by using the property of Laplace Transform known as

**Final Value Theorem**.

Definition of Final Value Theorem of Laplace Transform
If f(t) and f'(t) both are Laplace Transformable and sF(s) has no pole in jw axis and in the R.H.P. (Right half Plane) then,
Proof of Final Value Theorem of Laplace Transform
We know differentiation property of Laplace Transformation:
Note
Here the limit 0^{-} is taken to take care of the impulses present at t = 0
Now we take limit as s → 0. Then e^{-st} → 1 and the whole equation looks like

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Points to remember:

- For applying FVT we need to ensure that f(t) and f'(t) are transformable.
- We need to ensure that the Final Value exists. Final value doesn’t exist in the following cases

- Then apply

Answer Answer Note See here Inverse Laplace Transform is difficult in this case. Still we can find the Final Value through the Theorem.

Answer
Note
In Example 1 and 2 we have checked the conditions too but it satisfies them all. So we refrain ourselves of showing explicitly. But here the sF(s) has a pole on the R.H.P as the denominator have a positive root.
So, here we can’t apply **Final Value Theorem**.
Answer
Note
In this example sF(s) has poles on jw axis. +2i and -2i specifically.
So, here we can’t apply **Final Value Theorem** as well.
Answer
Note

In this example sF(s) has pole on the origin.
So here we can’t apply Final Value Theorem as well.
Final Trick
Just check that sF(s) is unbounded or not. If unbounded, then it is not fit for **Final Value Theorem** and the final value is simply infinite.

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