  This paper deals with convergence of a solution for the parabolic-parabolic
Keller-Segel system \[ (u_\lambda)_t = \Delta u_\lambda - \chi \nabla \cdot
(u_\lambda \nabla v_\lambda), \quad \lambda (v_\lambda)_t = \Delta v_\lambda -
v_\lambda + u_\lambda \quad \mbox{in} \ \Omega\times (0,\infty) \] to that for
the parabolic-elliptic Keller-Segel system \[ u_t = \Delta u - \chi \nabla
\cdot (u \nabla v), \quad 0= \Delta v -v +u \quad \mbox{in} \ \Omega\times
(0,\infty) \] as $\lambda \searrow 0$, where $\Omega$ is a bounded domain in
$\mathbb{R}^n$ ($n\ge 2$) with smooth boundary, $\chi, \lambda>0$ are
constants. In chemotaxis systems parabolic-elliptic systems often provided some
guide to methods and results for parabolic-parabolic systems. However, there
have not been rich results on the relation between parabolic-elliptic systems
and parabolic-parabolic systems. Namely, it still remains to analyze on the
following question except some cases: Does a solution of the
parabolic-parabolic system converge to that of the parabolic-elliptic system as
$\lambda \searrow 0$? In the case that $\Omega$ is the whole space
$\mathbb{R}^n$, or $\Omega$ is a bounded domain and $\chi$ is a strong signal
sensitivity, some positive answers were shown in the previous works. Therefore,
one can expect a positive answer to this question also in the Keller-Segel
system in a bounded domain $\Omega$ in some cases. This paper gives some
positive answer in the 2-dimensional and the higher-dimensional Keller-Segel
system.
