Which notation provides " a limit does not exist"?

Question

I was told by my teacher that it's not common to use a logical symbol like ∄ to show a limit nonexistence. But i was kind of unsure due to the exact definition of this notation which is "does not exist". So I'm looking for an authenticated source to prove my opinion. I was wondering if you know any:)

Answer 1

In the search for truth, you ought to be looking for all sources, whether or not they prove your opinion!

As someone who has taught calculus many times and who has studied analysis, I have never seen the existential quantifier used to denote a divergent limit. The symbol $\exists$ is a logical symbol, and the usual syntax is something like $\exists x.x\in\mathbb{R}\wedge x>0$ for the statement "there exists something which I will call $x$ such that $x$ is a real number and $x$ is positive." The $x$ is the variable bound by the existential quantifier, and it's the whole $\exists x$ that goes together. I have seen $\exists_x$, too. The symbol $\not\exists x.P(x)$ is shorthand for $\neg\exists x.P(x)$, "it is not the case that there exists an $x$ such that $P(x)$ is true."

As someone who has an informal personal notation for taking notes, I tend to use $\exists$ and $\not\exists$ in place of the words "exists" and "not exists." This is like how the $\in$ notation came about: it is a lunate variant of the Greek letter epsilon ($\varepsilon$), used to stand in for the first letter of the Latin word est, meaning "is." In fact, in very old books you can see things like $x\mathbin{\varepsilon} X$ for $x$ being an element of the set $X$. I might consider writing $$\lim_{x\to 0}x^{-1} = \not\exists,$$ but only in notes that are for my own eyes only. I would never write $\not\exists\lim_{x\to 0}x^{-1}$. It's hard to say exactly why because limit notation is actually fairly strange when you think about it: what $\lim_{x\to 0}f(x)=L$ means is that there exists a certain function that computes a $\delta$ for every $\varepsilon$ that gives a certain condition on how far $f(x)$ is from $L$ in a certain range of $x$ about $0$. What does $\lim_{x\to 0}f(x)$ mean on its own, without the $=$? Is it a statement that is true or false depending on whether the limit exists? Is it the value to which $f(x)$ limits if the limit exists? But then what if the value does not exist? is it just "undefined" there? Basically, the notation changes meaning depending on context, and it is up to the astute reader to figure out which one it means. The notation $\not\exists\lim_{x\to 0}x^{-1}$ makes it seem like somehow the question itself of whether or not that function has a limit at $0$ does not exist; at least that is what it suggests to me. Better notation might be the statement $$\operatorname*{nolim}_{x\to 0}x^{-1},$$ but, again, I'd never use it publicly (at least without sufficient explanation about what it is I mean!)

As someone who has been through the excitement of learning new notation: know that most of it is not used by professional mathematicians who are not logicians. When I write mathematical research papers, I will write out complete sentences, saving symbols for when their terseness can make things clearer. Really, $\therefore$, $\because$, $\forall$ and $\exists$ are pretty much not used, and rarely even at the board, save for some shorthands like "w.r.t." for "with respect to" and "TFAE" for "the following are equivalent." The implication arrow $\Rightarrow$ is often used, too, but it goes to show how little it is used in formal writing that I had to look the symbol up in Detexify. The feeling of writing an inscrutible alchemical work might be alluring, but a mathematical problem is not truly solved until you are able to communicate it to your peers. Unnecessary bespoke notation just gets in the way of this.

A professor told me yesterday that they had found that requiring all their students to write their answers in complete sentences, giving no points to those who don't, noticeably improved the quality of their students work. They thought it freed the students from thinking that math was somehow different from plain old clear and logical communication. It also saves students from digging themselves into the hole of trying to communicate using only "$=$" and "$\Rightarrow$" as verbs! Students who try pretty much always make a complete mess of things since they seem to want these symbols to mean more than they actually do. If a nonstandard use of a symbol takes explaining, just explain it and dispense with the symbol.

In short: (1) No, it is not common. (2) Math is not about trying to come up with the tersest notation to represent an idea, unless that somehow makes things easier to understand for others.

Answer 2

It is helpful to think about what we are doing with these logical symbols and why we use them. The goal of mathematics is not to write the most unreadable logical sequence imaginable. In fact, we are actively trying to communicate when we write mathematics. Otherwise, every statement would be boiled down to ZFC and nobody would be able to understand what anybody is saying. We use logical symbols when they help us be more explicit about what we are saying. This is not such a case and so it is not recommended to be used here.

Another thing is, in logic, there are specific rules about how to construct sentences and formulae. In your linked example

$$\lim_{x \to \infty} f \not \exists$$

This is not an admissible sentence. It could be "correctly" written as

$$\not \exists a \lim_{x \to \infty} f = a$$

or

$$\forall a \lim_{x \to \infty} f \neq a$$

but there's an important key there. In logic, we can only add quantifiers $\exists$ and $\forall$ if they are immediately followed by a variable and then a formula.

I do want to reiterate that even though it is logically sound to say $\forall a \lim_{x \to \infty} f \neq a$, you should ask yourself if this makes your mathematics easier to read. I would argue that it is much easier to read "$\lim_{x \to \infty} f$ does not exist."