Discourse; Limit; Realizations; Mathematics for Teaching; Math Routine. 

In this research work, we have developed a macro project on the discursive processes triggered

in the teaching and learning of the concept of limit, which was carried out through the

realization of three studies with particular objectives, which aim to broaden the understanding

about the complexities surrounding the discourse on limit and contribute to the research agenda

of this theme. In order to develop the studies, the following actions were carried out: a document

analysis with scientific articles on discursive conflicts in the understanding of the concept of

limit; a documentary analysis with scientific articles on the realizations of the concept of limit;

and a documentary analysis of two textbooks collections on mathematical routines. The first

study revealed the existence of the following discursive conflicts: the conflict of limit as an

approximation; the process-object conflict of limit; the conflict of continuity of limit; the

conflict of process-object duality of the symbol 𝑙𝑖𝑚𝑓(𝑥)=𝐿; and the conflict of the 𝑥→𝑎

inversibility of discourses x to y and y to x of the formal definition of limit. In the second study, we constructed a theoretical model of mathematics for the teaching of the concept of limit, based on the realizations on notion of limit which were documented in scientific articles which investigate the teaching and learning of this concept. The model was structured in four scenarios: limit as approximation; limit as a model to investigate infinity; limit as instantaneous rate of change; and limit as a formal mathematical object. For the third study, we analyzed two collections of mathematics textbooks of basic education in order to identify potentially useful mathematical routines in order to exploring the notion of limit at this level of education. It was identified ten potential routines, which were categorized as archethretic limit, algebraic limit and geometric limit. They developed three pedagogical imaginations based on the routines, one of each category. Taken together, these three studies provide a broad view on discourse of limits and can contribute in the fields of scientific production, teacher training and in the production of instructional materials aimed at teaching limits.