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 et al., 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 I(f,infinity) is computable directly from the experimentally measured waveform f(t) by an algorithm that is suitable for real-time calculation and implementation.