  Quantum duality principle is applied to study classical limits of quantum
algebras and groups. For a certain type of Hopf algebras the explicit procedure
to construct both classical limits is presented. The canonical forms of
quantized Lie-bialgebras are proved to be two-parametric varieties with two
classical limits called dual. When considered from the point of view of
quantized symmetries such varieties can have boundaries that are noncommutative
and noncocommutative. In this case the quantum duality and dual limits still
exist while instead of Lie bialgebra one has a pair of tangent vector fields.
The properties of these constructions called quantizations of Hopf pairs are
studied and illustrated on examples.
