"Limit" is a difficult mathematical concept, even for good students. It is also a key foundational idea for the study of advanced mathematics. It holds a central position as a foundation of the theory of approximation, of continuity, and of differential and integral calculus. However, the difficulty for teachers is how to organise teaching and learning. Using a constructivist approach, this study used a Teaching Experiment, supported by the use of Dynamic Geometry Software, to identify important steps for students in the development of the concept of a finite limit sequence. The results show that teaching using dynamic manipulations helps to identify shortcomings in students' thinking while challenging them to form and verify their hypotheses and conclusions about a finite limit.