  In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions
$u_p$ to $$ \left\{ \begin{array}{rclr} -\Delta_p
u_p&=&\frac{\lambda}{|x|^p}|u_p|^{p-2}u_p+f&\quad \mbox{ in } \Omega,\\ u_p&=&0
&\quad \mbox{ on }\partial\Omega, \end{array}\right. $$ where $\Omega$ is a
bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary,
$\lambda\in\mathbb{R}^+$, and $f$ is a nonnegative datum in
$L^{N,\infty}(\Omega)$. Under sharp smallness assumptions on the data $\lambda$
and $f$ we prove that $u_p$ converges to a suitable solution to the homogeneous
Dirichlet problem $$\left\{ \begin{array}{rclr}- \Delta_{1} u &=&
\frac{\lambda}{|x|}{\rm Sgn}(u)+f & \text{in}\, \Omega,\\ u&=&0 & \text{on}\
\partial \Omega,\end{array}\right. $$ where $\Delta_{1} u ={\rm
div}\left(\frac{D u}{|Du|}\right)$ is the $1$-Laplace operator. The main
assumptions are further discussed through explicit examples in order to show
their optimality.
