We present a didactic situation aimed at the formal definition (delta-epsilon) of the limit of a function; the experimentation of this didactic situation has been made many times with French students in the last year of high school and the first year of university using ''scientific debates'' between students. From an excerpt of the script of an experimentation, we study the evolution of students' reasoning. We specifically study the kinds of rationalities used by students to try to solve the given problem.