Fundamental studies on combustion in laminar, nonpremixed flames are often carried out using conserved scalar quantities. These conserved scalar quantities, represented here by mixture fraction, E, are used as independent variables in activation-energy asymptotic analysis and in rate-ratio asymptotic analysis. These analyses are carried out in the asymptotic limit of large Damköhler number, with chemical reactions presumed to take place in a thin reaction zone, that is located at E = Est. The quantity Est is the stoichiometric mixture fraction. A characteristic diffusion time is given by the reciprocal of the scalar dibipation rate, X. Previous computational studies have shown that the scalar dibipation rate at extinction depends on Est and the maximum flame temperature, Tst. Here a rate-ratio asymptotic analysis is carried out using reduced chemistry to elucidate the influence of Est on critical conditions of extinction. The scalar dibipation rate at extinction was predicted as a function of Est with the mab fractions of reactants so chosen that the adiabatic flame temperature, Tst, is fixed. The predictions of the analysis show that with increasing values of Est, the scalar dibipation rate at extinction first increases and then decreases. To test the predictions of the asymptotic analysis critical conditions of extinction are measured on nonpremixed methane flames stabilized in the counterflow configuration. With increasing values of stoichiometric mixture fraction, the strain rate at extinction was found to increase and the scalar dibipation rate at extinction was found to first increase and then decrease. The predictions of the asymptotic analysis agreed with experiments. A key outcome of the analysis is that with increasing Est the thickneb of the regions where oxygen and fuel are consumed first increase and the decrease. This is responsible for the observed non-monotonic changes in the values of the scalar dibipation rate at extinction with changes in Est.