Is there a generalization of limits s.t. $\lim_{n\to \infty}x_n$ and $\lim_{t\to 0}f(t)$ are special cases?

Question

Suppose that $x_n,f(t)\in[0,\infty[$ for all $n\in\mathbb N$ and $t\in\left]0,\infty\right[$. Note that the expressions $$\lim_{n\to \infty}x_n$$ and $$\lim_{t\to 0}f(t)$$ both makes sense (i.e. we know what it means that these limits exist). Can both expression be considered a special case of a definition in a more general setup/a more general type of domain which includes $\mathbb N$ and $]0,\infty[$ as special cases?

Edit 1: I know that we can define the convergence of sequences in topological spaces, but that is not my question.

Edit 2: Here is an attempt to make my question more precise: I think that one possible generalization would be to consider a triple $(D,F,f)$, where $D$ is a set, $F\subseteq P(D)$ and $f:D\to[0,\infty[$ some function. I guess that we may set $$\lim(D,F,f)=0\Leftrightarrow\forall\epsilon>0:\exists \emptyset\neq U\in F:\forall x\in U:f(x)\leq \epsilon.$$ and that this is also related to filters, which were mentioned in the comments. Can someone confirm this? Is this a standard definition?

Answer 1

You can define limits in any topological space! Just replace the epsilons and Deltas in the usual definition with a neighbourhood. In such a general settings weird things happen, for example in the space is not Hausdorff, the limit if it exists may not be unique

Answer 2

Yes indeed, as @LorenzoDeIaco said, you have topological spaces, but there is also a thing called filter theory which gives you the same level of generalization but the formalism is closer to your actual definition of limit.

Then there is a construction in category theory called (as we could expect) limits which is a generalization of limits in topological spaces (but it should be too advanced for you).

For you second edit, it is equivalent to a (badly written) limit for the discrete topology which is not very useful since everything converges with it: if you have some point $x$ where $f(x) = 0$ check what happens for $U = \{x\}$