We define the classical limit of quantum theory in the mathematical fr
amework of nonstandard analysis, choosing h as an infinitesimal number
. Up to corrections of infinitesimally small norm, bounded observables
which change continuously on the standard (non-infinitesimal) phase s
pace scale, are identified with functions on phase space. We discuss t
he convergence of commutators to Poisson brackets, and the quantum tim
e evolution to the classical one. These results are also shown for the
classical limit of spin systems, by choosing the spin as an infinite
half-integer.