The purpose of this study was to investigate the flow of thought of the pre-service mathematics teachers through the answers of a function limit evaluation by formal definition. This study used a qualitative approach with descriptive method. The research subjects were the students of mathematics education department of Universitas Serang Raya, Indonesia. After analyzing the students' written answers, we interviewed the subjects to get further explanation on their strategies and common mistakes. This study found that based on the students' results in the function limit evaluation by formal definition, there were common strategies, i.e. (1) preparing the proof and (2) proving. The stage of preparing the proof consisted of (1) determining delta value by the final statement of formal definition, (2) substituting the given f(x) and L process, (3) simplifying value in the absolute sign, (4) solving the inequality, and (5) finding the delta value. The stage of proving consisted of (1) stating positive epsilon, (2) defining delta, (3) stating positive delta, (4) substituting the constants and delta values in the initial statement of formal definition, and (5) solving the inequality to create the final inequality statement of the formal definition.