Difference Between Limit Point and Accumulation Point?
Basically an accumulation point has lots of the points in the series near it. A limit point has all (after some finite number) of 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.
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 $(x_n)_n \in N$.
I hope you can get some idea from this.
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.