Name for multi-valued analogue of a limit

Let the "multilimit" $L_{x \to v}f(x)$ denote a set of values that $f(x)$ can approach as $x$ approaches $v \in \mathbb{R}$ . More formally,

$$ \xi \in L_{x \to v}f(x) \stackrel{\text{def}}{\iff} \exists s \in \text{Seq}[\mathbb{R}] . \left(\lim_{t \to \infty} s_t = v \right) \land \left(\lim_{t\to\infty} f(s_t) = \xi\right) \tag{101} $$

This bizarre higher-order function of sorts, $L$, has the nice property of being total. The multilimit is a singleton if and only if the limit exists, but there's no harm in having an empty or non-singleton multilimit.

I'm curious whether there's an established usage of a construction like this.

A practical application to me seems to be defining a differential operator that always succeeds and can handle isolated poles (like in $\frac{1}{x}$) (102).

$$ D(f)(x) \stackrel{\text{def}}{=} L_{h \to v} \frac{f(x+h)-f(x-h)}{2h} \tag{102} $$

I've tried searching for something resembling a multivalued limit or for something resembling a multivalued generalization of the derviative but have only found the subderivative so far, which seems similar but doesn't directly tackle totality.

The most commonly used phrase for what you want is cluster set (see here also). See also the following Stack Exchange question/answers:

Cluster point of a function at a point

"Faint" continuity

Although the idea of the cluster set of a function at a point appears to have been considered by Cauchy (for example, p. 15 of Bradley/Sandifer's translation of Cauchy's Cours d'Analyse uses the notation $\lim \left(\left(\sin \frac{1}{x} \right)\right)$ for the cluster set of $\sin \frac{1}{x}$ at $x=0)$ and the idea is certainly in the background for issues related to the Casorati–Weierstrass theorem from the 1860s on (see also pp. 139-166 here), I believe the idea of the cluster set of a function at a point wasn't significantly studied or used until the 1890s, when Paul Painlevé applied this idea in the study of complex-valued functions (using the term domaine d'indétermination) and Rodolfo Bettazzi proved that each nonempty closed subset of the reals is the cluster set of some Baire one function at a point (it is easy to show that the cluster set of any function $f:\mathbb R \rightarrow \mathbb R$ is a nonempty closed subset of the extended real line). These are probably not the only two people who made use of this idea in the late 1800s, but (without my spending time looking into this issue) they're the only two who I know for sure did.

Until about 1960, the study and application of cluster set ideas followed two essentially disjoint paths, each pursued by people who were almost entirely unaware of the pursuits by those in the other path. One path was work done by William Henry Young (some of Young's work is described here), Henry Blumberg and his students, and a few others such as Stefan Jan Kempisty, Georges Louis Bouligand, Alexandru Froda, Fedor Isaakovich Shmidov, Frédéric Amédée Emile Roger, etc. Surveys of some of this work include On the Symmetric Structure of Unconditioned Point Sets and Real Functions by Herbert Charles Parrish (his 1955 Ph.D. dissertation, essentially under Henry Blumberg) and Cluster sets of arbitrary real functions: a partial survey by Charles Leonard Belna (1976).

The other path, in which there were many more publications, dealt with complex-valued functions, and some of the more prominent researchers are Wladimir Seidel, Joseph Leo Doob, Alberto Pedro Calderón, Fritz Herzog, Frederick Otto Bagemihl, Edward Foyle Collingwood, Kiyoshi Noshiro, etc. Surveys of some of this work include the two books Cluster Sets by Kiyoshi Noshiro (1960) and The Theory of Cluster Sets by Collingwood/Lohwater (1966), and the 1969 Ed.D. dissertation An Introduction to Cluster Set Theory by James Reid Calhoun.

Probably the “golden era” for research involving cluster sets and their applications, if I had to pick one, was during the 1960s and 1970s. Many publications during this period took up generalizations to metric, uniform, and topological spaces, and many results in real and complex analysis were generalized to various cluster set settings (e.g. the limit operator can be generalized to allow the neglect of negligible sets; in dimensions greater than $1$ limits can be taken radially or tangentially or by some other approach along a line or curve to a boundary point, or limits can be taken within a specified sector with vertex at the limiting point; and other possibilities). In fact, Theodore John Kaczynski’s mathematical research involves these kinds of concepts.

The specific application you asked about is to the subsequent limits of a symmetric difference quotient. The terms you want to search for are derived number or contingent or derivate interval (for the ordinary derivative version), and symmetric derived number (for the symmetric derivative version; I’m not getting many google hits for this, however).

In the past few decades there have been a very large number of papers on Set Valued Analysis in which one can find (among other topics) results about, and applications to, cluster sets of various difference quotient notions, for functions from and to various types of spaces. For example, I have a book on this subject and there are several journals specifically devoted to this field. The literature for this is so huge that unless you have a very specific question and there’s someone here who is reasonably knowledgeable about this field, I would advise just googling appropriate phrases, such as “symmetric difference quotient” + “set valued”.

Having been ignorant of the contents of Dave L Renfro's informative answer, I would have called it a multi-limit.