  The standard notion of a classical limit, represented schematically by
$\hbar\rightarrow 0$, provides a method for approximating a quantum system by a
classical one. In this work we explain why the standard classical limit fails
when applied to subsystems, and show how one may resolve this by explicitly
modelling the decoherence of a subsystem by its environment. Denoting the
decoherence time $\tau$, we demonstrate that a double scaling limit in which
$\hbar \rightarrow 0$ and $\tau \rightarrow 0$ such that the ratio $E_f =\hbar
/\tau$ remains fixed leads to an irreversible open-system evolution with
well-defined classical and quantum subsystems. The main technical result is
showing that, for arbitrary Hamiltonians, the generators of partial versions of
the Wigner, Husimi and Glauber-Sudarshan quasiprobability distributions may all
be mapped in the above double scaling limit to the same completely-positive
classical-quantum generator. This provides a regime in which one can study
effective and consistent classical-quantum dynamics.
