Is it correct that the concept $\lim _{x \to a} f(x)$ first requires that $a$ is a limit point of the domain?

Question

My textbook mentions the limit of a function as follows:

and remarks:

The authors define the limit point as follows:

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My questions:

From the textbook, I think that the concept $\lim _{x \to a} f(x)$ first requires there is at least a sequence $(x_k)$ in $D$ converging to $a$.

Could you please verify if my understanding is correct or not? Thank you so much!

Answer 1

Yes, you are right. Indeed, if $a$ is a limit point of $D$, then, for every $n\in\mathbb N$, there is some $x_n\in B_{1/n}(a)\cap D$. And then clearly $\lim_{n\to\infty}x_n=a$. Besides the fact that, of course, $(\forall n\in\mathbb N):x_n\in D$.