# Do points have limits?

1. May 22, 2015

### Blue and green

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

2. May 22, 2015

### Blue and green

Possibly kindly answer it as well. Thanks.
- blue

3. May 22, 2015

### FactChecker

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

4. May 23, 2015

### Fredrik

Staff Emeritus
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.

5. May 23, 2015

### 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 $x$).

6. May 24, 2015

### HallsofIvy

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

7. Jun 7, 2015

### disregardthat

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

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.

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,...)

8. Jun 12, 2015

### 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