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”The limits of category theory are a great generalization of an analogy with the limits discussed here.
It turns out, however, that limits in topological spaces (at least) can be viewed as category-theoretic limits.
For now, see this math.sx answer.（下記）”
（参考）
https://ncatlab.org/nlab/show/convergence
convergence
Redirected from "limit of a sequence".
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3. Properties
Relation to limits in the sense of category theory
The limits of category theory are a great generalization of an analogy with the limits discussed here.
It turns out, however, that limits in topological spaces (at least) can be viewed as category-theoretic limits.
For now, see this math.sx answer.（下記）

https://math.stackexchange.com/questions/60590/category-theoretic-limit-related-to-topological-limit/62800#62800
Category-theoretic limit related to topological limit?

This question came to me after I saw ( http://www.youtube.com/watch?v=be7rx29eMr4 ) a surprising fact that generalised metric spaces can be seen as categories enriched over preorder ([0,∞],=<).
asked Aug 29 '11 at 22:44 Rafael Mr?en

2 Answers
The connection is well-known (in particular I'm claiming no originality; I don't recall where I found this, though !):
略 answered Sep 8 '11 at 10:53 bonnbaki

https://ncatlab.org/nlab/show/limit
limit
This entry is about the notion of “limit” in category theory.
For the notion of the same name in analysis and topology see at limit of a sequence.