The amount of high-quality material on elementary calculus available for free online these days is an embarrassment of riches, so most of my suggestions for reading are online. I’ll refer to books in this section only by the surname of the first author; the references section below tells you where to find the book online or in print.
The reader who wants to learn more about the hyperreal system might want to start with Stroyan and the Mathforum.org article. For more depth, one could next read the relevant parts of Keisler. The standard (difficult) treatise on the subject is Robinson.
Given sufficient ingenuity, it’s possible to develop a surprisingly large amount of the machinery of calculus without using limits orinfinitesimals. Two examples of such treatments that are freely available online are Marsden and Livshits. Marsden gives a geometrical definition of the derivative similar to the one used in ch. 1 of this book, but in my opinion his efforts to develop a sufficient body of techniques without limits or infinitesimals end up bogging down in complicated formulations that have the same flavor as the Weierstrass definition of the limit and are just as complicated. Livshits treats differentiation of rational functions as division of functions.
Tall gives an interesting construction of a number system that is smaller than the hyperreals, but easier to construct explicitly, and sufficient to handle calculus involving analytic functions.
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