I want to clarify the definition of limit point and accumulation point.
According to many of my text books they are synonymous that is $x$ is a limit/accumulation point of set $A$ if open ball $B(x, r)$ contains an an element of $A$ distinct from $x$.
But from one of the problems in Aksoy: A Problem Book in Real Analysis says:
Show that if $x \in (M,d)$ is an accumulation point of $A$, then $x$ is a limit point of $A$. Is the converse true?
So what is the definition?
Basically an accumulation point has lots of the points in the series near it. A limit point has all (after some finite number) of the points near it.
Think of the series $(-1+\frac 1{n^3})^n$. Both $-1$ and $1$ are accumulation points as there are entries very far out close to each. Neither is a limit because there are points very far out that are far away.
In my undergrad real analysis class, a limit point is defined as such:
Let E $\subseteq R$ and x $\in R$. Then x is a limit point of E if x is the limit of a sequence of points in E.
i.e. $\exists$ {$x_n$} in E such that $x_n \to x$ as $n \to \infty$.
Using this definition of limit point and your definition of accumulation point, it is possible for a point to be a limit point of a set but not an accumulation point of that same set.
e.g. The singleton set E={1} contains a sequence of constant terms, {1,1,1,...} which converges to 1. But 1 is not an accumulation point of E because there is no open neighborhood of 1 that contains an element of E distinct from 1.
However, using these definitions, it can be shown that every accumulation point of a set is a limit point of the same set.
From the Wikipedia article on "accumulation points":
Accumulation point is a type of limit point.
types of limit points are:
if every open set containing x contains infinitely many points of S then x is a specific type of limit point called an ω-accumulation point of S.
If every open set containing x contains uncountably many points of S then x is a specific type of limit point called a condensation point of S.
If every open set U containing x satisfies $|U ∩ S| = |S|$ then x is a specific type of limit point called a complete accumulation point of S.
A point $x \in X$ is a cluster point or accumulation point of a sequence $(x_n)_n \in N$ if, for every neighbourhood V of x, there are infinitely many natural numbers $n$ such that $x_n \in V$.This is equivalent to the assertion that $x$ is a limit of some subsequence of the sequence $(x_n)_n \in N$.
According to your description, I suppose by "being a limit point of a set $S$" the author means "being a point $x$ such that every open ball of center $x$ contains at least one point of S", which is usually referred to as an adherent point (see Apostol's Mathematical Analysis). Then by "being an accumulation point of $S$" the author must mean "being a point $x$ such that every open ball of center $x$ contains at least one point of $S$ distinct from $x$" (i.e. being an adherent point of the set $S\setminus \{ x \}$). The term "accumulation point" is a common one (see Apostol's Mathematical Analysis).
Using the terms employed by Askoy, it is clear that every accumulation point of $S$ is a limit point of $S$. This simply follows from the definitions; being an accumulation point of $S$ requires more than being a limit point of $S$. The opposite is not necessarily true; consider the set $\{1 \}$, of which $1$ itself is a limit point but not an accumulation point by definition.
First giving the definition of an 'adherent' point : A point $x$ is said to be adherent to the set $S$ if an open set centered at $x$ contains some element of $S$. Thus an adherent point can be a boundary point or an interior point of $S$.
Now, in particular an accumulation point is adherent to $S$. Thus it can lie either in the set or can be a boundary point. Whereas a limit point lies on the boundary.
This is very crude but you can get the idea.
Thus what Ross Millikan says is true.
I think they are the same, there only exists difference between closure points and accumulation points or limit points.
You can refer to http://mathworld.wolfram.com/AccumulationPoint.html for more details. What's more, the limit point is also called cluster point.
From taking an undergraduate Real Analysis course, I believe the answer depends on what kind of space our sequence is in.
If we're in a metric space, I believe this can be more complicated and a limit point is NOT the same thing as an accumulation point. I think there is at least one distinction between a limit point and an accumulation point inside of metric spaces, namely that they both describe conditions for a different number of distinct, other points inside neighborhoods or balls around the limit/accumulation point.
If we had a neighborhood around the point we're considering (say $x$), a limit point's neighborhood would be contain $x$ but not necessarily other points of a sequence in the space, but an accumulation point would have infinitely many more sequence members, distinct, inside this neighborhood as well aside from just the limit point. In other words, accumulation points describe a "build up" of similar, distinct points nearby and not just the convergence trend of a sequence of points (inside a metric space, anyway).
Some set theory notation describing what I think the difference is for some sequence of points in a subset $E$ of the metric space $\{x_n\}$ that converges to $x$ (assuming $x$ is also in $E$ and we're simply using $\varepsilon > 0$ as our all radius, and using $d$ as our metric space function):
An open ball around $x$ as a limit point would mean: $$B_\varepsilon(x, d)\cap E≠\varnothing$$
An open ball around $x$ as an accumulation point would mean: $$B_\varepsilon(x, d)\setminus\{x\}\cap E≠\varnothing$$
Relevance of Distinction
As far as I know, the relevance is that in metric spaces, not all Cauchy sequences are convergent -- all Cauchy sequences being convergent means a metric space is "complete", which is a property related to "compact", i.e., the differences between the types of convergence points can be used to help illustrate what sorts of sequences in a given space we have and whether or not it is Cauchy or convergent.
Hope this helps!
Do not confuse the limit of a sequence with the limit points of a set.
As already pointed out by others, a limit point x of a set A is defined by this feature: the intersection of any neighborhood of x with A always contains at least one point different from x. The point x itself may or may not belong to A.
For all the topologies coming from metric spaces, and more generally for each Hausdorff topology, being a limit point implies being also an accumulation point. But in other kinds of topologies (e.g., a finite set with at least two points, with the the trivial topology) this is not always true: every accumulation point of a set is also a limit point, but limit points may not be accumulation points.
See, for instance, the comment here: https://encyclopediaofmath.org/wiki/Condensation_point
