The Wikipedia article on the Final Value Theorem states the following for cases where it does not hold:
There are two checks performed in Control theory which confirm valid
results for the Final Value Theorem:
1. All roots of the denominator of H(s) must have negative real parts.
2. H(s) must not have more than one pole at the origin.
The first condition makes sense, since the time domain limit will not exist, but what is the reason for the second condition?
Best Answer
For example, if we have a function with two poles in the origin, as: $$F(s)=\frac{1}{s^2}$$ We obtain: $$f(t)=t$$ That, for $t \to \infty$, is obviously infinity.
You can also demonstrate it solving the limit: $$\lim_{s\to0}s \frac{1}{s^2}$$