Limit and limit points What is the basic difference between limit and limit points, and if a sequence has one unique limit how it can have a lot of limit points 
 A: A limit point is a generalization of a limit (each limit is a limit point but not vice versa). You can see this in the definition:
Limit: $a$ is a limit of $(a_n)$, iff in each neighborhood of $a$ are almost all elements of $(a_n)$.
Limit point: $a$ is a limit of $(a_n)$, iff in each neighborhood of $a$ are infinite elements of $(a_n)$.
I highlighted the difference between both definitions.
Because only infinite elements need to be in any neighborhood of $a$. There can be infinite elements outside this neighborhood which can clustered to another limit point. This is the reason why there might be many limit points for a sequence. If $a$ is a limit only finite elements can be outside any given neighborhood which is not enough to go to another limit.
A: Limit points are also often referred to as “cluster points” or “accumulation
points,” but the phrase “$x$ is a limit point of some set $A$” has the advantage of explicitly reminding us that $x$ is quite literally the limit of a sequence in $A$.
In fact, a point $x$ is a limit point of a set $A$ if and only if $x = \lim a_n$ for some sequence $(a_n)$ contained in $A$ as long as $a_n \neq x$ for all natural $n$.
Does this help in any way?
