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.
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An infinitesimal is a positive number whose absolute value is less than any assignable positive number. Observe the word assignable. The difference between assignable and non-assignable  was well understood by the founders of the calculus. It is arrogant to think that the giants like Newton, Lebniz and Euler
did not know that there is no nonzero number whose absolute value is less than any positive real. They bear in mind the fact that there are two kinds of reals. Our concept of standard  is the ancient assignable in a modern logical disguise. 
When one asks for an example of infinitesimal  which is the inverse of an infinite and says there are no such, he  mocks himself for the numbers are mental constructions, and there are no numbers independent of human beings. That is, he cannot give an example of any number in much the same way as  he knows no "real" examples of infinitely large and infinitely  small numbers. 
If we think about the numbers expressing the quantity of molecules in a room or if we count the grains of sands on a beach, then we see that these numbers are unreachable  in contract to the number of fingers. The natural series starts with assignable numbers but there are clearly some  numbers that are practically unreachable by successive count. These examples explain the difference between standard or assignable and non-standard or non-assignable numbers on a naïve level.
