Summary: We are interested in the numerical approximation of the complex Ginzburg-Landau equation in the large time and space limit. There are two interesting regimes in this problem, one is the large space time limit, and the other is the nonlinear Schrödinger limit. These limits have been studied analytically [see, for example, T. Colin and A. Soyeur, Asymptotic Anal. 13, No. 4, 361–372 (1996; Zbl 0885.35127); F. H. Lin, Commun. Pure Appl. Math. 51, No. 4, 0385–0441 (1998; Zbl 0932.35121); F. H. Lin and J. X. Xin, Commun. Math. Phys. 200, No. 2, 249–274 (1999; Zbl 0920.35145)]. We study a time splitting spectral method for this problem. In particular, we are interested in whether such a scheme is asymptotic preserving (AP) with respect to these two limits. Our results show that the scheme is AP for the first limit but not the second one. For the large space time limit, our numerical experiments show that the scheme can capture the correct physical behavior without resolving the small scale dynamics, even for transitional problems, where small and large scales coexist.