Good evening. I'm going to ask a question which might appear dumb or worse, but I consider myself both a student, a researcher and a continuous learner and there are times in which even the tiniest things have to be properly sift. Especially for a Mathematician!
Few years ago I was teaching about limits and most of all about directional limits. Now, I am a very strong fan and supporter of the Heine method with limits, for I have always wanted to sterilise the thinking of the students from the left-right dichotomy (and to prepare them to what will happen in $\mathbb{R}^n$). But at the times I just taught directional limits in the usual way (Cauchy criterion and so on).
Now, here is the point: when dealing with directional limits at finite points, especially when calculating them at the boundary of a function's domain ($\mathrm{e.g.}\quad \Omega: x\in (-\infty, -2)\cup(-2, 7)\cup(7, +\infty)$) I used to explain them both via a graphic way but also via the "so called" $\epsilon-$ method.
I have to say that I never found this method in any book I have read, but I am rather sure it is not something that strange or uknown. What is this method about? It's just a translation of what, say, $a^+$ or $a^-$ means, in terms of infinitesimals. In this way we write
$$\lim_{x\to 2^+} f(x) \equiv \lim_{\substack{x\to 2 + \epsilon \\\\ \epsilon > 0 \\\\ \epsilon \to 0}} f(x) $$
One example
$$f(x) = \exp\left(\dfrac{3x+3}{x^2-3x}\right)$$
The domain reads $\Omega: x\in(-\infty, 0) \cup (0, 3) \cup (3, +\infty)$. Say we want to calculate the limit at $3$, with no graphic help and no other method. Then for what concerns the exponent:
$$\lim_{x\to 3^+} \dfrac{3x+3}{x^2-3x} \equiv \lim_{\substack{x\to 3 + \epsilon \\\\ \epsilon > 0 \\\\ \epsilon \to 0}} \dfrac{3x+3}{x^2-3x}$$
And now we directly substitute, and evaluate the limit at the end:
$$\lim_{\substack{x\to 3 + \epsilon \\\\ \epsilon > 0 \\\\ \epsilon \to 0}} \frac{9 + 3\epsilon + 3}{9 + \epsilon^2 + 6\epsilon - 9 - 3\epsilon} = \lim_{\substack{x\to 3 + \epsilon \\\\ \epsilon > 0 \\\\ \epsilon \to 0}} \frac{12 + 3\epsilon}{\epsilon^2 + 3\epsilon} \longrightarrow +\infty$$
Whence $$\lim_{x\to 3^+} \exp\left(\frac{3x+3}{x^2-3x}\right) \to e^{+\infty} \to +\infty $$
Where it was easy to see the behaviour, since the numerator goes like $12$ and the denominator is a positive infinitesimal quantity.
In the same way, we can obtain
$$\lim_{x\to 3^-} \exp\left(\frac{3x+3}{x^2-3x}\right) \equiv \lim_{\substack{x\to 3 - \epsilon \\\\ \epsilon > 0 \\\\ \epsilon \to 0}}\exp\left(\frac{3x+3}{x^2-3x}\right) = \lim_{\substack{x\to 3 - \epsilon \\\\ \epsilon > 0 \\\\ \epsilon \to 0}} \exp\left( \frac{12-3\epsilon}{\epsilon^2 - 3\epsilon}\right)$$
The numerator goes like $12$ again. Here we see that since $\epsilon >0$ but infinitesimal, then $\epsilon^2 < \epsilon$ and $\epsilon^2 < 3\epsilon$. Thence $\epsilon^2 - 3\epsilon$ is a negative infinitesimal quantity (or we can neglect $\epsilon^2$ and remain just with $-3\epsilon$) and this term goes to $-\infty$ namely:
$$\sim \lim_{\substack{x\to 3 - \epsilon \\\\ \epsilon > 0 \\\\ \epsilon \to 0}} \exp\left( \frac{12}{- 3\epsilon}\right) \to e^{-\infty} \to 0$$
In a very simple and similar way one can deal with limits at $0$.
The positive side of this, is that we indeed work with infintesimals, which are always positive ($\epsilon > 0,\ \epsilon \to 0$) so there is no sign ambiguity.
Now, finally, the question : is there some reason why this method couldn't or shouldn't be used? As I said before, I never found this in any book I have ever read, and this made me think that it might be not a valid or suitable method for directional limits. So I appeal to you who can give me more information, kindly excusing possible ill-notations that I may have used in this question.
Thank you so much!