Final Value Theorem (FVT)

Learned in MATH213.

Theorem 3: Final Value Theorem

For a function $F(s)$, if

• is a proper rational function
• has the property that all the poles have real parts that are strictly negative with the exception of a single pole (of order 1), i.e. $\frac{A}{s}$ (NOT $\frac{A}{s^2}$), at $s=0$

or if $F(s)$ is the product of a function satisfying the above conditions multiplied by a complex exponential $e^{sT}$ , then ${\lim }_{t\to \infty }f(t)={\lim }_{s\to 0}sF(s)$

If the poles of a rational function do not satisfy the above condition, then ${\lim }_{t\to \infty }f(t)$ does not exist.

If the F does not satisfy those two conditions...

Then you say that the limit does not exist.

Related

• Initial Value Theorem