In this article, we provide further asymptotic analysis to the higher-order statistics (HOS) of the channel capacity over generalized fading channels, especially by proposing simple and closed-form expressions each of which can be easily computed as a tight-bound revealing the existence of constant gap between the actual and asymptotic HOS of the channel capacity in the limit of both high- and low-signal to noise ratios (SNRs). As such, we show that these closed-form asymptotic expressions are insightful enough to comprehend the diversity gains. The mathematical formalism we followed in this article is illustrated with some selected numerical examples that validate the correctness of our newly derived asymptotic results.