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At the start of the higher education curriculum, the conceptualization oflocal approximation objects of a function requires the articulation ofknowledge and skills from Functional Analysis and Topology. In the study offunctions, a number of studies have established the existence of difficultiesencountered by students, mainly as a result of the change of didactic contractduring the transition from secondary to higher education. The construction of ateaching-learning project, as part of a collaborative effort with the classteacher, a priori helps students to overcome the main difficulties inherent inconceptualizing the local approximation objects of a function in the first yearof preparatory classes. In the case of the design and implementation ofdidactic development engineering, analysis of the reasoning produced bystudents confronted with a situation with an adidactic dimension will enable usa priori to study the nature and origin of these difficulties. Our methodologyfor analyzing student work is based on a model of reasoning analysis within theframework of the theory of didactic situations mathematics. This model hasplayed an essential role in the development of didactic engineering, in theidentification of students' conceptions, forms and functions of reasoning. Italso enabled us to identify epistemological, didactic and cultural obstacles tolearning the concept of local approximation of a function. These obstaclesresult either from the paradigm shift that takes place during the transitionfrom secondary to higher education, or from working within the paradigm ofInfinitesimal Analysis during the appropriation of this mathematical concept.

Mise en {\oe}uvre d'une ing\'enierie didactique de d\'eveloppement dans  le cadre d'un travail collaboratif chercheur/enseignant lors de la  conceptualisation des objets de l'Analyse au d\'ebut du cursus dans le  sup\'erieur