Reasoning about geometric limits

Continuing our critique of the classical derivation of the formula for the area of a disk, we focus on the limiting processes in geometry. Evidence suggests that intuitive approaches in arguing about infinity, when geometric configurations are involved, are inadequate, and could easily lead to erroneous conclusions. We expose weaknesses and misconceptions in arguments which attempt to articulate what the static geometric object ‘in the limit’, or ‘at infinity’, is, or what it looks like. Supported by school curricular expectations and existing research, we suggest an alternative focus and reasoning process to teaching and investigating geometric limits. Instead of discussing what happens ‘at infinity’, or ‘at the end of a limiting process’, we suggest that one should argue based on what happens as we move ‘far enough’, i.e. ‘far toward infinity’. Thus, we suggest replacing the static situation ‘at infinity’ with the investigation of the corresponding dynamic geometric pattern. This way, the emphasis is placed on reasoning about approximations and estimations, which appear as objectives in school curricula of many countries. We encourage introducing and discussing geometric limits, as they raise a number of interesting questions, ask for identification of patterns, and have potentially unexpected and exciting outcomes.