Remark about scaling limit of ABJ theory

We generalize the suggestion presented in arXiv:0806.3498 that the 3d N = 8 superconformal SU(N) Chern-Simons-matter theory of Lorentzian Bagger-Lambert- Gustavson type (L-BLG) can be obtained through the scaling limit from N = 6 superconformal U(N) x U(N) Chern-Simons-matter theory of Aharony, Bergman, Jafferis and Maldacena (ABJM) to the case when we study the scaling limit of N = 6 superconformal U(M) x U(N) Chern-Simons-matter theory of Aharony, Bergman and Jafferis (ABJ). We show that if we extend the ABJ theory in the same way as in arXiv:0811.1540 we can define a consistent limit that leads to SU(N) L-BLG theory together with U(M-N)(k) Chern-Simons theory of level k.