  The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra
$H$ is studied. Such a Hopf algebra $H$ is an extension of an affine
commutative Hopf algebra $A$ by a finite dimensional Hopf algebra $F$. The main
theorem gives natural conditions under which $H^{\circ}$ decomposes as a
crossed or smash product of $F^{\ast}$ by the finite dual of $A$. This
decomposition is then further analysed using the Cartier- Gabriel-Kostant
theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the
classical components of $A^{\circ}$. The detailed consequences for a number of
families of examples are then studied.
