Over the last forty years of mathematics education research, a coherent body of knowledge has accumulated regarding the teaching of limits. On this basis, it remains a challenge to identify goals and design tasks compatible with ordinary teaching conditions. This paper reports on a teaching experiment carried out in France with year 12 students, which led to the formulation "by the students" of a correct formal definition of the infinite limit for sequences, with minimal background logical prerequisites and in the course of a 2-h session. On a more theoretical level, the teaching project was developed in the framework of didactic engineering, and provides opportunities to contribute to the ongoing work on its adaptation to the specific context of tertiary education. In the a priori analysis, we highlight the didactical potential of tasks of "differentiation" between neighboring concepts as a pathway to advanced mathematical concepts. In the a posteriori analysis, we focus on the nature and extent of teacher intervention in the shaping of a mathematical "milieu" that is conducive to the definition of an advanced mathematical concept.