Is there a name encompassing both limit inferior and limit superior

Question

Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?)

The only thing I was able to find was that some authors use the name extreme limits; see google books: "extreme limits" "lim sup". It is used e.g. in Thomson, Bruckner, Bruckner: Elementary real analysis, p.70

EDIT: I've checked a few more searches to see to which extent the above term is widespread. Searching for "extreme limits" only gives a lot of non-mathematical result, so these are basically attempts to filter out results that are interesting for this question.

• Google Books: "extreme limits" sequence
• Google Books: "extreme limits" function
• Google books: "Subject:Mathematics" "extreme limits"
• Google Scholar - searching in Engineering, Computer Science, and Mathematics: "extreme limits"

NOTE: This question arose more or less out of curiosity. When I discussed limit superior and limit inferior with some other MSE users, we found out that none of us was familiar with a name which would describe these two things. (Of course, it might be the case that such term is not needed very much, if we are only able to find something which is used frequently.)

Answer 1

It seems that the term extreme limits is used, although not very frequently.

Some people call limit superior and limit inferior inner and outer limits, see e.g. this spikedmath comic strip (Wayback Machine). This is not a single-word name, but it was interesting for me, since I did not know it before.

Some examples of usage of this usage terms can be found here (although many non-mathematical uses appear there, too):

Extreme limits

• Google Books: "extreme limits" sequence
• Google Books: "extreme limits" function
• Google Scholar - searching in Engineering, Computer Science, and Mathematics: "extreme limits"

Inner and outer

• Google Books: inner "outer limits" sequence
• Google Books: inner "outer limits" function
• Google Scholar - searching in Engineering, Computer Science, and Mathematics: inner "outer limits"

Answer 2

I would go with cluster point, as the target $\mathbb R$ is a metric space. See this PlanetMath link.

Answer 3

For what it is worth I do not think there is any well-established terminology for what you want. I have never heard of any, anyway.

I think it is worthwhile to change your question to a slightly broader question: "what kinds of common generalizations could one make of these ideas, so that they become different instances of the same general thing, and what terminology might one use for the generalization?"

A question like has more than one right answer. Depending on what properties you regard as essential to any abstraction, and what your perspective is, you will get different results. (Even something more fundamental than all of these limitlike operations, namely the completeness property of the set of real numbers, takes on very different "flavors" depending on your point of view: is it an order-theoretic property, or a metric property, or what?)

Here is one view of the issue based on a "dynamical" sort of perspective: you might justifiably call each of $\liminf$, $\limsup$, and $\lim$ "asymptotic properties" of a sequence. Let me formalize a definition here.

Consider the map $\sigma$ on the set of all sequences of real numbers defined by sending the sequence $(x_n)_{n=1}^{\infty}$ to $(x_{n+1})_{n=1}^{\infty}$ (what many call "the shift").

The natural domains of $\liminf$, $\limsup$, and $\lim$ are all invariant under $\sigma$ (by "natural domain" I just mean "the largest set of sequences on which the operation can be defined and gives a real result", which in the cases of $\liminf$, $\limsup$, and $\lim$ are the sets of bounded below, bounded above, and convergent sequences, respectively). By this I mean that if a sequence $x$ is in the domain of any one of these operations, then $\sigma(x)$ is also in that domain. And of course each of the operations $\liminf$, $\limsup$, $\lim$ is invariant under $\sigma$, by which I mean that the result of applying the operation to a sequence $x$ in its domain is the same as the result of applying the operation to $\sigma(x)$.

Here's a first approximation to a general definition:

Suppose that $\mathcal{D}$ is a subset of the set of all real sequences with the property that $\sigma(\mathcal{D}) \subseteq \mathcal{D}$. Then a function $F: \mathcal{D} \to \mathbb{R}$ is called an asymptotic property of sequences in $\mathcal{D}$ if $F(\sigma(x)) = F(x)$ holds for all $x$ in $\mathcal{D}$.

By this definition $\liminf$ and $\limsup$ and $\lim$ are all asymptotic properties of sequences in their domains, whereas e.g. $\inf$ or $\sup$ or "evaluate at the 200th term", in their natural domains, are not.

The intuitive reason for choosing the word "asymptotic" is hopefully clear: if $F$ has the property given in the definition, then although the number $F(x)$ depends on $x$, it does not depend on any finite number of terms taken from $x$. So in a vague sense it depends only on what is happening "way out there".

(To make this notion slightly more reasonable, you might require that the domain $\mathcal{D}$ be more than just a shift-invariant set of sequences. To avoid trivialities it seems natural to require that $\mathcal{D}$ contain, or even properly contain, the set of constant sequences, and also to require that $F$ coincide with "evaluation at the first entry" on the set of constant sequences.)

More structured versions of this "asymptotic property" concept do exist and have names. For example: require $\mathcal{D}$ to be the set of all bounded sequences, regarded as a topological vector space with the componentwise vector space operations and the topology coming from the supremum norm, and require $F$ to be a linear map that is continuous with respect to this topology, and require additionally that $F$ coincide with $\lim$ on the set of convergent sequences (and perhaps also add a positivity condition), and you have what is called a Banach limit. But $\limsup$ and $\liminf$, not being linear, are not instances of this concept.