In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if ${\displaystyle f(t)}$ in continuous time has (unilateral) Laplace transform ${\displaystyle F(s)}$, then a final value theorem establishes conditions under which $${\displaystyle \lim _{t\,\to \,\infty }f(t)=\lim _{s\,\to \,0}{sF(s)}.}$$ Likewise, if ${\displaystyle f[k]}$ in discrete time has (unilateral) Z-transform ${\displaystyle F(z)}$, then a final value theorem establishes conditions under which $${\displaystyle \lim _{k\,\to \,\infty }f[k]=\lim _{z\,\to \,1}{(z-1)F(z)}.}$$

An Abelian final value theorem makes assumptions about the time-domain behavior of ${\displaystyle f(t){\text{ (or }}f[k])}$ to calculate ${\textstyle \lim _{s\,\to \,0}{sF(s)}.}$ Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of ${\displaystyle F(s)}$ to calculate ${\displaystyle \lim _{t\to \infty }f(t)}$ ${\displaystyle {\text{(or }}\lim _{k\to \infty }f[k])}$ (see Abelian and Tauberian theorems for integral transforms).

In the following statements, the notation ${\displaystyle {\text{‘}}s\to 0{\text{’}}}$ means that ${\displaystyle s}$ approaches 0, whereas ${\displaystyle {\text{‘}}s\downarrow 0{\text{’}}}$ means that ${\displaystyle s}$ approaches 0 through the positive numbers.

Suppose that every pole of ${\displaystyle F(s)}$ is either in the open left half plane or at the origin, and that ${\displaystyle F(s)}$ has at most a single pole at the origin. Then ${\displaystyle \lim _{t\to \infty }f(t)=\lim _{s\to 0}sF(s).}$

Suppose that ${\displaystyle f(t)}$ and ${\displaystyle f'(t)}$ both have Laplace transforms that exist for all ${\displaystyle s>0.}$ If ${\displaystyle \lim _{t\to \infty }f(t)}$ exists and ${\displaystyle \lim _{s\,\to \,0}{sF(s)}}$ exists then ${\displaystyle \lim _{t\to \infty }f(t)=\lim _{s\,\to \,0}{sF(s)}.}$

Remark

Both limits must exist for the theorem to hold. For example, if ${\displaystyle f(t)=\sin(t)}$ then ${\displaystyle \lim _{t\to \infty }f(t)}$ does not exist, but $${\displaystyle \lim _{s\,\to \,0}{sF(s)}=\lim _{s\,\to \,0}{\frac {s}{s^{2}+1}}=0.}$$

Suppose that ${\displaystyle f:(0,\infty )\to \mathbb {C} }$ is bounded and differentiable, and that ${\displaystyle tf'(t)}$ is also bounded on ${\displaystyle (0,\infty )}$. If ${\displaystyle sF(s)\to L\in \mathbb {C} }$ as ${\displaystyle s\to 0}$ then ${\displaystyle \lim _{t\to \infty }f(t)=L.}$