We study traveling wave solutions for the class of scalar reaction–diffusion equations ∂ u ∂ t = ∂ 2 u ∂ x 2 + f m ( u ) , where the family of potential functions { f m } is given by f m ( u ) = 2 u m ( 1 − u ) . For each m ⩾ 1 real, there is a critical wave speed c crit ( m ) that separates waves of exponential structure from those which decay only algebraically. We derive a rigorous asymptotic expansion for c crit ( m ) in the limit as m → ∞ . This expansion also seems to provide a useful approximation to c crit ( m ) over a wide range of m-values. Moreover, we prove that c crit ( m ) is C ∞ -smooth as a function of m −1 . Our analysis relies on geometric singular perturbation theory, as well as on the blow-up technique, and confirms the results obtained by means of asymptotic methods in [D.J. Needham, A.N. Barnes, Reaction–diffusion and phase waves occurring in a class of scalar reaction–diffusion equations, Nonlinearity 12 (1) (1999) 41–58; T.P. Witelski, K. Ono, T.J. Kaper, Critical wave speeds for a family of scalar reaction–diffusion equations, Appl. Math. Lett. 14 (1) (2001) 65–73].