Is $\epsilon^2/\epsilon^2=1$ or $0/0$? Is it possible in the system of dual numbers ($a+\epsilon b$; $\epsilon^2=0$) to calculate $\epsilon/\epsilon =1$? How then does one deal with $\epsilon^2/\epsilon^2=1$ versus $\epsilon^2/\epsilon^2=0/0$?
The same question for infitesimal calculus using hyperreal numbers where: $\epsilon \neq 0$ but $\epsilon^2=0$?
I probably did not use the correct formulation w.r.t. hyperreal numbers. I meant the axiom (?) in smooth infinitesimal analysis where it is assumed: $\epsilon \neq 0$ but $\epsilon^2=0$.
I am not quite sure how this analysis is related to nonstandard-analysis and hypercomplex numbers. I came across this topic in the book: A Primer of infinitesimal analysis (John L. Bell).

As far as smooth infinitesimal analysis (SIA) is concerned, it is not correct that it assumes that $\epsilon \neq 0$.  It cannot be proven that $\epsilon \neq 0$.  What is true is that infinitesimals in SIA satisfy $\neg\neg(\epsilon = 0)$.
