  This paper investigates the zero relaxation limit for general linear
hyperbolic relaxation systems and establishes the asymptotic convergence of
slow variables under the unimprovable weakest stability condition, akin to the
Lax equivalence theorem for hyperbolic relaxation approximations. Despite
potential high oscillations, the convergence of macroscopic variables is
established in the strong $L^\infty_t L^2_x$ sense rather than the sense of
weak convergence, time averaging, or ensemble averaging.
