The more precise definition of differential $dx$ when defining $dy=f'(x)\Delta x$? [Edit] I think I made a logical mistake here and this question is flat out false, Please don't waste your time to give more answers.

This might be a stupid question, but I'm kind of lost in the notations here.
After reading some articles and textbooks (for instance https://encyclopediaofmath.org/wiki/Differential), I am start wondering if there is a better definition for $dx$
Given a differentiable function $y=f(x)$, I have
$$\Delta y=f'(x)\Delta x + o(x)$$
Then the differential $dy$ can be defined as
$$dy=f'(x)\Delta x$$
It seems then the differential $dx$ can be defined as:
$$dx=\Delta x$$
So far so good, but then I am suppose to have
$$\lim\limits_{\Delta x \to 0}\frac{\Delta y}{\Delta x}=\frac{dy}{dx}=\frac{f'(x)\Delta x}{dx}=\frac{f'(x)dx}{dx}=f'(x)$$
And during the middle of it I can get
$$\lim\limits_{\Delta x \to 0}\Delta x=dx=0 \ (????)$$
What mistakes have I made, and is there an equation that can define $dx$ more precisely just like $dy$ ?
 A: The modern approach does not use infinitesimals, precisely because of such apparent inconsistencies: a positive infinitesimal is smaller than any positive real number, but not zero. The issue is that such a real number doesn't exist, so infinitesimals aren't real numbers. But all the limit laws are usually developed for real numbers only, so how are we supposed to work with infinitesimals when calculating limits, which is at the heart of calculus?
There are two solutions to this: either introduce a rigorous way to treat infinitesimals (this is called non-standard analysis, and I suppose most mathematicians never bother to dive deep into this theory), or just don't use infinitesimals. The latter is the standard approach.
In this approach, the symbols $\mathrm dx$ and $\mathrm dy$ don't have any inherent meaning. They are only meaningful in combinations like $\frac{\mathrm dy}{\mathrm dx}$ or $\int f(x)\mathrm dx$. Just like $\lim$ by itself has no meaning, but $\lim\limits_{x\to a}f(x)$ has meaning, or the fraction bar $\frac{~}{~}$ has no meaning without a numerator and denominator. Here, $\frac{\mathrm df}{\mathrm dx}:=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. Full stop. No further explanations what the numerator and denominator in that expression mean, since they don't mean anything. Only the full expression carries meaning, and it can't be manipulated like a fraction. In particular, things like $f'(x)\mathrm dx$ have no meaning and shouldn't come up in a rigorous treatment.
There's a caveat: In differential geometry, we have things called differential forms, which we write as $\mathrm df$ or $\mathrm dx$. But these don't have a whole lot to do with what you probably imagine when thinking of infinitesimals, and probably noone calls them infinitesimals.
