  Bundles of C*-algebras can be used to represent limits of physical theories
whose algebraic structure depends on the value of a parameter. The primary
example is the $\hbar\to 0$ limit of the C*-algebras of physical quantities in
quantum theories, represented in the framework of strict deformation
quantization. In this paper, we understand such limiting procedures in terms of
the extension of a bundle of C*-algebras to some limiting value of a parameter.
We prove existence and uniqueness results for such extensions. Moreover, we
show that such extensions are functorial for the C*-product, dynamical
automorphisms, and the Lie bracket (in the $\hbar\to 0$ case) on the fiber
C*-algebras.
