  In this paper, we first give a lower bound of the lifespan and some estimates
of classical solutions to the Cauchy problem for general quasi-linear
hyperbolic systems, whose characteristic fields are not weakly linearly
degenerate and the inhomogeneous terms satisfy Kong's matching condition. After
that, we investigate the lifespan of the classical solution to the Cauchy
problem and give a sharp limit formula. In this paper, we only require that the
initial data are sufficiently small in the $L^1$ sense and the BV sense.
