From Process Control: A First Course with MATLAB
2.4. Initial and FinalValue Theorems
Two theorems are now presented that can be used to find the values of the timedomain function at two extremes, t = 0 and t = ?, without having to do the inverse transform. In control, we use the finalvalue theorem quite often. The initialvalue theorem is less useful. As we have seen from our first example in Section 2.1, the problems that we solve are defined to have exclusively zero initial conditions.

InitialValue Theorem:
(2.23) 
FinalValue Theorem:
(2.24)
The finalvalue theorem is valid provided that a finalvalue exists. The proofs of these theorems are straightforward. We will do the one for the finalvalue theorem. The proof of the initialvalue theorem is in the Review Problems.
Consider the definition of the Laplace transform of a derivative. If we take the limit as s approaches zero, we find
If the infinite integral exists, ^{[9]} we can interchange the limit and the integration on the LHS to give
Now if we equate the RHSs of the previous two steps, we have
We arrive at the finalvalue theorem after we cancel the f(0) terms on both sides.
Example 2.1:
Consider the Laplace transform F( s) = {[6( s  2)( s + 2)] /[ s( s + 1) ( s + 3)( s + 4)]}. What is f( t = ?)?
Example 2.2:
Consider the Laplace transform F
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