Previously a new method for ultrasound signal characterization using entropy H f was reported, and it was demonstrated that in certain settings, further improvements in signal characterization could be obtained by generalizing to Renyi entropy-based signal characterization I f (r) with values of r near 2 (specifically r=1.99) [M. S. Hughes, J. Acoust. Soc. Am. 125, 3141-3145 (2009)]. It was speculated that further improvements in sensitivity might be realized at the limit r→2. At that time, such investigation was not feasible due to excessive computational time required to calculate I f (r) near this limit. In this paper, an asymptotic expression for the limiting behavior of I f (r) as r→2 is derived and used to present results analogous to those obtained with I f (1.99). Moreover, the limiting form If,∞ is computable directly from the experimentally measured waveform f (t) by an algorithm that is suitable for real-time calculation and implementation.