We investigate the transfer function of the discretized perfect lens in finite-difference time-domain (FDTD) and transfer matrix method (TMM) simulations; the latter allow to eliminate the problems associated with the explicit time dependence in FDTD simulations. We also find that the finite discretization mesh acts like imaginary deviations from $\mu =\epsilon =-1$ and leads to a crossover in the transfer function from constance to exponential decay around $k_{\parallel ,\max }$ limiting the attainable super-resolution. We propose a simple qualitative model to describe the impact of the discretization. $k_{\parallel ,\max }$ is found to depend logarithmically on the mesh constant in qualitative agreement with the TMM simulations.