  Let $\Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic
upper half plane $\mathbb H$, and let $M = \Gamma \backslash \mathbb H$ be the
associated finite volume hyperbolic Riemann surface. If $\gamma$ is parabolic,
there is an associated (parabolic) Eisenstein series, which, by now, is a
classical part of mathematical literature. If $\gamma$ is hyperbolic, then,
following ideas due to Kudla-Millson, there is a corresponding hyperbolic
Eisenstein series. In this article, we study the limiting behavior of parabolic
and hyperbolic Eisenstein series on a degenerating family of finite volume
hyperbolic Riemann surfaces. In particular, we prove the following result. If
$\gamma \in \Gamma$ corresponds to a degenerating hyperbolic element, then a
multiple of the associated hyperbolic Eisenstein series converges to parabolic
Eisenstein series on the limit surface.
