arXiv:1701.05187math.HO
From Pythagoreans and Weierstrassians to true infinitesimal calculus
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Abstract
In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting "the definition" to the students as a monolithic absolute. We hope that this could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicates that students tend to favor infinitesimal definitions over epsilon, delta ones.
Here's an interesting paper on the topic for epsilon-delta proofs: https://arxiv.org/pdf/1701.05187 Also see Table 2 of this older paper for limit thinking in general: https://u.math.biu.ac.il/~katzmik/sullivan76.pdf
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