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A new dielectric slab waveguide with double-negative (DNG) film layer and single-negative (SNG) cover layer is proposed. The dispersion relations of an asymmetric left-handed materials (LHM) slab waveguide are investigated by using normalized parameters. An analytical method is then proposed to calculate the dispersion curves. TE-polarized guided modes differ dramatically from TM-polarized guided modes. For TE-polarized oscillating guided modes, fundamental modes exist all the time, but in a narrow range of low frequency values only. First order mode behaves as other higher order modes and exists up to infinite high frequency. Higher order modes have double degeneracy. For TE-polarized surface guided modes, the existence and the type of the mode solutions with respect to different parameters are classified systematically and discussed in detail. TE0 surface mode exists only in one case. The dispersion curve of TE1 surface mode continues with that of oscillating TE0 mode. It seems that the two different kinds of modes compensate each other to form one whole mode. For TM-polarized wave, both oscillating and surface guided modes act as the waveguide with DNG (LHM) film sandwiched in between two normal dielectrics.