Abstract: Stolz theorem is an important tool in the mathematical analysis to solve the limit of 3 0 and 3 ∞ type. Many scholars use differentmethods to p rove the Stolz theorem, and from different directions to generalize a series of important research results. In this paper, we set up a necessary and sufficient condition of the inverse p roposition of the Stolz theorem, generalize it to the function form, and solve some p roblems. The conclusions of this paper is the further generalization of the Stolz theorem.