Asymptotic comparisons of ergodic channel capacity at high SNR are provided for optimal power and rate adaptation and channel inversion over fading channels with general distributions, as well as the capacity of the equivalent AWGN channel. We present closed-form expressions for the high-SNR capacity difference gaps among these capacities. Based on these expressions it is shown that the presence of space diversity or multi-user diversity make channel inversion arbitrarily close to achieving optimal capacity at high SNR with sufficiently large number of antennas or users. In addition, for the case of low SNR, it is shown that a variant of channel inversion can be made close to the optimal scheme, and provide better capacity under fading than the equivalent AWGN channel. Numerical results are shown to corroborate our analytical results.