Yes. All the points of the interior of the circle are limit points in the circle. However, not all the limit points of the circle are in the interior of the circle.

Specifically, in , the closed circle where is the center and r is the radius contains all of its limit points.

Your notation is hard to understand. I've never seem Lim(A). It may mean the set of all limit points?

In a more specific example,

let A={ } where A is a subset of

let Lim(A) be the set of all limit points of A (I am assumingthis is what Lim(A) means btw)

is Lim(A)={ } where Lim(A) is a subset of ?

but then the closure of A, Cl(A)=Lim(A) which makes me question my understanding of the definition of limit points...or is that right?

Intuitively, I think of Lim(A)={0} union { + =1} which is actually Bd(A)

The closure of A is different from the set of all its limit points because it may contain its isolated points as well. In other words, the closure of A is the union of A and its limit points.