  We consider families of degenerating hyperbolic surfaces. The surfaces are
geometrically finite of fixed topological type. Let Z(s) be the Selberg Zeta
function of a surface, and let Z_d(s) be the contribution of the pinched
geodesics to the Zeta function. Extending a result of Hejhal and Wolpert, we
prove that the quotient of these two terms converges to the Zeta function of
the limit surface for all arguments s with re(s)>1/2. The technique is an
examination of resolvent of the Laplacian, which is composed from that for
elementary surfaces via meromorphic Fredholm theory. The resolvent is shown to
converge on the complement of the essential spectrum of the limit surface. We
also use this property to define approximate Eisenstein functions and
scattering matrices.
