Why does the definition of a limit of a function not appear in textbooks on general topology? Surely, continuity is a very basic topological concept.  But I don’t recall any explicit definition of limit of a function as we find in calculus.
I think it is incorrect to mark this question as a duplicate.  The cited other question asks for sources of where limit has been defined in topology.  I ask why limit of a function is not usually defined.  Indeed, Noah’s answer addresses this question.
 A: The usual notion of limit generalizes naturally to arbitrary topological spaces: given spaces $(X,\tau)$ and $(Y,\sigma)$ and a map $f:X\rightarrow Y$, we can say $\lim_{x\rightarrow a}f(x)=b$ (with respect to $\tau$ and $\sigma$) iff $$\forall U\in\sigma\exists V\in\tau(b\in U\implies V\not=\emptyset\wedge\forall x\in V\setminus\{a\}(f(x)\in U)),$$ or in words iff for every open set $U$ in the target containing $b$ there is some open set $V$ in the source containing $a$ which $f$ sends entirely inside $U$ (with the possible exception of $a$ itself). It's not hard to check that this agrees with the usual notion of continuity in analysis, and for reasonably nice spaces it's single-valued so actually makes good sense (to see the potential issue, consider the indiscrete topology; it's a good exercise to pin down what "reasonably nice" means in this context).
So why don't limits appear more centrally in a basic point-set topology class (at least in the context of spaces in which they are guaranteed to be unique when they exist)? 
Well, first of all they do, at least partly: limits of sequences are in fact things we care about rather frequently. It is true, though, that we talk much less (at least initially) about limits at specific points of functions between general topological spaces, focusing more on the question of overall continuity. So the question still remains.
I think the situation is best explained by recalling why we care about limits in classical analysis in the first place: we're trying to give a formal treatment of calculus, and the key notions of calculus - derivatives and integrals - both correspond to taking limits of appropriate functions at appropriate points. But the apparatus of calculus does not generalize well to arbitrary topological spaces, and so local questions like "What, if anything, is $\lim_{x\rightarrow a}f(x)$?" wind up being less important (at least initially) than global questions like "Is the map $f$ continuous?" The limit idea becomes primarily a tool for answering such questions - e.g. to show that a function isn't continuous we find some point where its limit either doesn't exist or differs from its actual value. But we can develop such arguments without using the limit language.
Of course that's a subjective explanation which many might reasonably disagree with, but I think it's more-or-less accurate. That said, I do think that introductory topology classes should indeed present this generalization if only as an example of how to use topological ideas (I did when I taught it, and if I recall correctly Munkres' book does as well).
