This paper introduces infinite intervals into the Duration Calculus.
The extended calculus defines a state duration 
over an infinite interval by a property which specifies
the limit of the state duration 
over finite intervals, and excludes the description operator. 
Thus the calculus can be established without
involvement of unpleasant calculation of infinity. With limits of
state durations, one can treat conventional liveness and
fairness, and can also measure out liveness and fairness
by properties of limits. Including both finite and infinite
intervals, the calculus can, in a simple manner, distinguish between
terminate behaviour and nonterminate behaviour, and therefore 
directly specify and reason about sequentiality.