In order to realize an electroweak first-order phase transition, a category of extended Higgs models with relatively large self-coupling constants is often considered. In such a scenario, the running coupling constants can blow up at an energy scale much below the Planck scale. To clarify the allowed parameter space of the model, it is important to evaluate the scale where perturbative calculation breaks down. In the renormalization group equation analysis using the mass-independent renormalization scheme, we need a matching condition to connect the low energy theory and the high energy theory. Consequently, the analysis depends on the details of the matching condition. On the other hand, an analysis with the mass-dependent beta function can be performed in a simpler way because the threshold effect is automatically included in the beta function. In this paper, using the mass-dependent beta function, we discuss the application limit of perturbative calculations at the one-loop level. First, we explain the essence of our method and compare our results with those based on the mass-independent beta function in a toy model with two complex scalar fields. We then apply our method to a more realistic model, i.e., the inert doublet model. We find that, in our method, in which the threshold effect is automatically included, the energy scale where the perturbative calculation breaks down can be higher than the one using the mass-independent beta function, especially when the blowup scale is relatively low. Published by the American Physical Society 2024