When the variation range of attribute independent variables is large in the Multinomial Logit (MNL) model, using the point elasticity value at the original value of the attribute independent variable to approximate the actual elasticity leads to a non-negligible first-order remainder error in the predicted change range of the choice probability. To address this issue, the first-order Lagrange remainder of the actual elasticity is derived based on the first-order Taylor formula of the mode choice probability function, and the maximum value and the extremum point of the absolute value of the first-order remainder are obtained. Then, given the error limit of the remainder-point elasticity ratio, within the neighborhood of the original value of the attribute independent variable, with the constraint that the upper limit of the absolute value of the first-order remainder-point elasticity ratio does not exceed the error limit, the analytical form of the applicable neighborhood for predicting the change range of the choice probability is systematically derived when using the own- and cross-point elasticity values to approximate the actual elasticity, which fully ensures the prediction accuracy of the change range of the choice probability. Finally, through numerical experiments, by taking three alternative modes of bus, car, and bicycle, and by using the bus fare as the attribute independent variable, the correctness of the theoretical derivation of the applicable neighborhood is verified.