Is there a limit definition of a differential? I came up with this but I would like some feed back.
\begin{align*} \text{d}x & = \lim_{x \to c}(c - x)\\ \text{d}x & = \lim_{\Delta x \to 0} \Delta x \end{align*}
It's just the denominator of the limit def of a derivative but is that legit?
Also, does $\text{d}x$ stand for "The Differential of x" or is it just short hand for "Delta x"
On
To be precise, differential is also known as an infinitesimal. It's a hyperreal number. As the link suggests, it's not a real number. But all reals are hyperreals. A satisfying explanation of infinitesimals was given by Robinson in $1970$'s. Non standard analysis deals with the study of infinitesimals. Non standard analysis explains the infinitesimal construction via compactness theorem and it's role in modern math. Non standard analysis even explains derivatives and integrals in much more intuitive way then using limits. For instance,
Derivatives:- In a calculus course it's taught that general definition of a derivative is, $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ Which is slope at a point. Now it doesn't seem to make sense that slope at a point exists but since the limit at $0$ converges and exists we have that it does have a slope at a point. But non standard analysis defines derivative as, $$f'(x)=st\left(\frac{f(x+dx)-f(x)}{dx}\right)$$ Where $st(..)$ is the standard part function. It basically is a function which takes every hyperreal to it's nearest real.
Despite the fact that non standard analysis is a better way to show the beauty behind any concept(involving infinitesimals) it's not feasible to learn it as it requires a lot of basics to be learnt. It's like as i heard from somewhere It's like a lot of pain for a little gain
Usually when you do a rigorous and more formal treatment of calculus, there is no such thing as $\mathrm{d}x$. It is nothing more than a symbol to remind you that you are working infinitesimally. Take for example the $\mathrm{d}x$ that might appear in an integral. You would probably write something like
$$\int_a^b f(x)~\mathrm{d}x$$
for your integral. But this $\mathrm{d}x$ is just a symbol to remind you if how this integral is actually defined, and how you can intuitively think of it. Leaving out the exact detail you would have something like
$$\int_a^b f(x)~\mathrm{d}x=\lim_{N\to\infty}\sum_{j=0}^N f(x_j^*)~\Delta x_j.$$
And here you might think "but doesn't $\Delta x_j\to\mathrm{d}x$ in the limit?", and while this is somewhat suggestive of something like this, you have to know that this is a limit of the entire sum at once, and not just this $\Delta x_j$, and so there is ko $\mathrm{d}x$ just magically coming into existence. A similar thing can be said about the $\mathrm{d}x$ that you see in the derivative, as there you have the suggestive definition
$$\frac{\mathrm{d}f(x)}{\mathrm{d}x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}.$$
In the end, though, you have to keep in mind that even though this is a fraction before you take the limit, it doesn't really stay that way.
Now as a little sidenote, you could still talk about $\mathrm{d}x$ if you introduce differential forms, but that's a rather separate thing from this that I don't think is all thay relevant to your question at the moment.