What's an example of an infinitesimal? If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give to the students? What I am asking for are specific techniques for explaining infinitesimals to students, geometrically, algebraically, or analytically.
Note 1. This page is related as is this.

My pedagogical answer is to explain one over a generic natural number.
We cannot explicitly write down a generic natural number just as we cannot explicitly write a generic (non-constructible) irrational number. Like a non-constructible irrational number, it is an abstraction. We do know that it is larger than any fixed integer. We have no algorithmic method to determine any of its non-trivial properties, such as whether it is even or odd, prime or composite. Indeed, we have no algorithmic method to distinguish two different generic natural numbers.
I feel that this approach is close to the infinitesimals of old, and it's also highly intuitive. The notion of one over a generic natural number as an "example of an infinitesimal" comes from  Kauffman's version of Sergeyev's grossone. It also relates to a view I have heard Tim Gowers express online, that a large integer out to be judged by how much we can say about it, and therefore (my words now) that one over a generic natural number is "functionally" an infinitesimal quantity.  
