Let $f:\mathbb R \to \mathbb R$ be a function and $a\in \mathbb R$ a point. The Cauchy definition of the limit $\lim _{x\to a}f(x)=L$ is well-known. For pedagogical reasons I'm interesting in a comprehensive list of other, equivalent, formalisms. For instance, the Heine concept of limit: if $x_n$ converges to $a$, then $f(x_n)$ converges to $L$, is an equivalent formalism to the Cauchy definition, one that relies on the notion of limit of a sequence.
At least two other formalisms I can think of are 1) using the hyperreals (i.e., $f(x)$ is infinitesimally close to $L$ whenever $x$ is infinitesimally close to $a$) and 2) using filters (i.e., the filter generated by the direct image of a filter converging to $a$, converges to $L$) both relying on some set-theoretic constructions.
Any other formalism that can be added to the four above will be greatly appreciated. Thanks.