I am currently working on a project for school where I would like to create an arithmetic framework that would make it easier to solve limits. Suppose we have a function $f:\mathbb{R}\smallsetminus \{a\} \to \mathbb{R}$, where we would like to evaluate $\displaystyle\lim_{x \to a^{+}} [f(x)]$ or $\displaystyle\lim_{x \to a^{-}} [f(x)]$. If $a^+$ or $a^-$ were numbers, informally "infinitesimally larger or smaller than $a$", they would be part of the domain for the functions and as such, you could just plug them in. That would be quite convenient. Additionally, I would like to propose that the distributive property applies to these infinitesimals. Although I have read about similar things before (non-standard analysis, hyperreals), I have yet to see a strictly defined framework for such numbers.
The first reasonable step on this journey is to accept the existance of a positive infinitesimal, i.e. a "number" $\varepsilon$ with the following quality: $\forall \delta\in \mathbb{R}^{+}:0<\varepsilon<\delta$. This seems axiomatic in that it simply has to be accepted for anything else to make sense. Now based on this definition, it seems certain limits also have to be changed a bit, specifically, limits that would traditionally approach 0 would have to approach this number instead:
$\displaystyle \lim_{n \to 0^{\pm}} n=\pm\varepsilon\approx 0$.
Some other interesting consequences of this definition seem to be that: $k\cdot \varepsilon=\varepsilon$ and $\varepsilon^k=\varepsilon$.
It seems you can work out some basic things like this without much issue. However, there seems to be situations where things begin to break down. For example, a very convenient consequence of the definition above is: $\displaystyle\lim_{x \to 0^{+}} \frac{f(x)}{x}=\frac{f(\varepsilon)}{\varepsilon}=\infty$, for some strictly positive function $f$. But we also know that $k\cdot \varepsilon=\varepsilon$. So what happens in a situation such as:
$\displaystyle \frac{\varepsilon}{\varepsilon}$
It seems to me this should be $1$, but maybe that would lead to some contradiction down the road. That segways out of the lengthy introduction into my main question: when constructing such a framework, how can we make sure it will not lead to contradictions down the road? Mainly, I would like to use these "numbers" to solve limits, so obviously the number system has to be defined in such a way where all kinds of limits evaluate to the same thing they would in standard analysis. Also, what kind of axioms/postulations would I need to "prove" some of the qualtities for the infinitesimal that I showed here? "Proof by intuition" isn't very rigorous...
Leibniz's great insight was that when one introduces infinitesimals in a number system, they should behave in the "same way" as "ordinary" numbers do. Thus, if $\epsilon>0$ is infinitesimal then $k\epsilon > \epsilon$ for $k>1$ and $\epsilon^k <\epsilon$, etc. This enabled him to develop a workable system of infinitesimals that can be used in analysis.
The idea of infinitesimals of different order was widely discussed by Leibniz. At wiki, the only related discussion I have been able to find is Cauchy's Cours d'Analyse.