Can Calculus/Analysis Help Determine Point Limits?

[Blue and green]

Can someone rephrase the title question into something more meaningful in terms of Calculus/Analysis?

 

[Blue and green]

Possibly kindly answer it as well. Thanks.
- blue

 

[FactChecker]

A single point does not but a series of points can approach a limit.

 

[Fredrik]

Suppose that $x_1,x_2,\dots$ is a sequence of points. A point $x$ is said to be a limit of that sequence if every open neighborhood of $x$ (i.e. every open set that contains $x$) contains all but a finite number of the points in the sequence.

 

[pwsnafu]

There is a difference between the individual point $x$ and the sequence $(x,x,x,\ldots)$. The former the does not have a limit but the latter does (and the limit is $x$).

 

[HallsofIvy]

Suppose that $x_1,x_2,\dots$ is a sequence of points. A point $x$ is said to be a limit of that sequence if every open neighborhood of $x$ (i.e. every open set that contains $x$) contains all but a finite number of the points in the sequence.

Just to clarify, here we would say that x is the limit, not that x "has" a limit!

 

[disregardthat]

Note that sequences may have many limits. If your topological space is not hausdorff, this may happen.

Suppose that $x_1,x_2,\dots$ is a sequence of points. A point $x$ is said to be a limit of that sequence if every open neighborhood of $x$ (i.e. every open set that contains $x$) contains all but a finite number of the points in the sequence.

This is slightly imprecise depending on interpretation. It should either say all but a finite number of terms in the sequence, since the sequence might eventually stabilize.

There is a difference between the individual point $x$ and the sequence $(x,x,x,\ldots)$. The former the does not have a limit but the latter does (and the limit is $x$).

Not sure how to interpret this, but I'd point out that the limit points of the one-point set $\{x\}$ are exactly the limit points of the sequence (x,x,...)

 

[Stephen Tashi]

The terms "limit of a sequence of points" and "limit point of set of points" have different meanings. A set of points (not necessarily arranged as a sequence) can have many "limit points". https://en.wikipedia.org/wiki/Limit_point