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I'm thinking about teaching calculus by firstly introducing the asymptotic notations (big-Oh, little-oh, and $\sim$), secondly explaining their "arithmetic" (things like how to sum little-oh's and similar), and then doing everything else (for example, the derivative of $f$ in $x_0$ would be defined as the real number $f^\prime(x_0)$, if it exists, such that $f(x) = f(x_0) + f^\prime(x_0) (x - x_0) + o(x - x_0)$ for $x \to x_0$).
Note that, of course, while explaining little-oh's I would implicitly explain the definition of limit to $0$.
In my experience, asymptotic notations are usually introduced to students after most of the theory have been shown, and mostly as a tool to compute limits. So I wonder if this alternative approach have been tried before and could be fruitful. Some pro/cons I see:
Asymptotics notations give a concise language to express many concepts ("bounded" is $O(1)$, "infinitesimal" is $o(1)$, the expression for the derivative that I already mentioned...) and work with them in an almost mechanical way.
Students have to learn them anyway soon or later
- Asymptotic notations could be a bit difficult to understand at first, because they are not reflexive (things like $o(x^2) = o(x)$ but $o(x) \ne o(x^2)$ for $x \to 0$).