Just check the definition of continuity for instance: If dx is not zero, what do you make of this? So to cure those inconsistencies, the epsilon-delta prescription was introduced, and is used today. For instance, the derivative of x 2 is (x+dx) 2-x2 divided by dx, which is 2x+dx. In case you don't know about the epsilon-delta prescription, the story is: Newton and Leibniz used infinitesimals, but soon mathematicians realized that mathematics is inconsistent when using them. So I don't think the nonstandard analysis is a way to go. This can be circumvented by the nonstandard analysis, which postulates the existence of infinitesimals, instead of using the epsilon-delta prescription, but the nonstandard analysis is 1) hard to envision in an intuitive way, and 2) it's not free from inconsistencies either, and 3) actually can lead one to the results different from the results of the standard analysis. So wither way, infinitesimals or epsilons, fundamental questions cannot be answered, but put forward inconsistencies. Yes, all this is about the "for every epsilon there is delta". And when they don't obey the usual arithmetic - they break mathematics. And they cannot possible obey the usual arithmetic. The main object there are cardinal numbers, which are infinite. For instance, the very first issues came about when Georg Cantor was working on his Set Theory. There are also issues with non-infinitesimals in standard mathematics. Yes, there are issues with infinitesimals. However, I'm under impression you're not interested in nonstandard, but in standard analysis. There's nonstandard analysis, there's link in other post. The idea of some infinitely small change however is mathematically sloppy and leads to logical problems. You can look at them as nilpotent elements of commutative rings (that is objects who are not necessary equal to zero but if you multiply them with themselves give zero) for example.
In modern mathematics, there are different ways to rigorously define objects that work similar to infinitesimals. It has however nothing to do with infinitely small quantities, that notion has been abandoned by mathematicians for almost 200 years. This construct is sometimes useful as it provides us with a neat linear approximation of f(x) that is easy to memorize because it agrees with the leibniz notation of derivatives. Both dy and dx are real numbers (not infinitesimals!!!) and we can manipulate them as such. Here, dy is itself a function of two separate, independent real variables f'(x) and dx. They are related to our function f by the equation dy=f'(x)dx. Now, as we like to have linear approximations of functions, we define ourselves quantities dx and dy which are named like that for historical reasons but are mathematically a totally different concept from infinitesimals. dx is not the limit of delta x for delta x tending to zero (that limit would obviously be zero). You can't meaningfully define something like an infinitesimal in standard analysis (at least not in the intuitive sense that newton and leibniz used them as positive numbers that are infinitely small), it is a flawed concept introduced by people 350 years ago because they had no real understanding of limits then.
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