Is there a connection between limit point of a subset of a metric space and the limit of a function?

Question

Is there a connection between limit point of a subset of a metric space and the limit of a function, or limit of a sequence?

I am not sure but I don't think there is because there can be more than one limit point of a subset of a metric space whereas the limit of a function is unique.

Is there a connection between these two terms that I am missing?

Answer 1

Let $(X,d)$ is a metric space and $S \subseteq X$. Then $x$ is a limit point of $S$ if and only if there exists a sequence $(x_j)_{j=0}^\infty$ in $S$ such that $x_j \to x$.

Answer 2

The most direct connection would probably be "Suppose that a sequence $(x_n)$ is in a metric space $X$ and $x_n\to x$, then $x$ is the limit point of $S=\{x_n|n\in \Bbb N\}$ iff $S$ is an infinite set."

This means that the limit of a sequence is also the only limit point of the range of the sequence whenever the sequence is not eventually constant.