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The wave propagation in an adiabatic tapered waveguide with left-handed metamaterial (LHM) core and air cladding is investigated by mode-matching method. The results show that the forward guided modes will always couple to the backward modes at the 'critical' thickness in the lossless waveguide. So, the propagating wave will not be completely standstill at the 'critical' thickness as proposed in previous reports, but be 'bounced' back. Besides, the propagation of wave in a lossy LHM tapered waveguide is also investigated.