What is a limit point?

Question

I'm new in calculus and can't understand what the limit point is. Here the definition from the textbook that I've a question about:

Let $E \subseteq \mathbb{R}$ and $a \in \mathbb{R}$. Then $a$ is a limit point of $E$ if $$ \forall \delta \in \mathbb{R}^+ : ( (a - \delta, a + \delta) - \{a\} ) \cap E \neq \emptyset. $$

(Original image here.)

1. Suppose that $E = (1,2)$. It says that $1$ and $2$ are also limit points, but I can't understand fully why? Or is it only because of the definition?

2. In 1., $1$ and $2$ are limit points, so $1.5$ too from the picture. Why do I have to extract $\{a\}$ if in the end it's also limit point?

3. If $E = (1,2) \cup (3,5)$ then what's the value of $\delta$, or should I answer it separately?

4. What is the real use of limit point?

5. Let $E = \mathbb{R}$, so every point in the real numbers is a limit point. How to use it?

It's seem stupid but I'm really curious about it. Please Help! -Thank you =w=

Answer 1

The definition says that $a$ is a limit point if $E$ contains points "infinitely close" to $a$ (in better words, as close as you want, as expressed by $\forall\,\delta>0$). The point $a$ is also said to adhere to the set $E$.

It should be obvious to you that any point of a real interval is a limit point, as well as both endpoints, even if they are not included in the interval.

Limit points are used to "complement" the open intervals, and are very useful for example to extend functions that are undefined at certain points.

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In particular, an extremely important application is to compute the slope of a curve at a point by considering the slopes of the chords to nearby points.

Extremely extremely important and useful.

Answer 2

Well, this confusion is common when anyone reads about "limit points" for the first time. To clear the confusion, let me first define something called as an "open ball" about a certain point.

So, let $x \in \mathbb{R}$ be a point. An "open ball" of radius $\delta$ is defined as the following set

$$B \left( x, \delta \right) = \left\lbrace y \in \mathbb{R} | d \left( x, y \right) < \delta \right\rbrace$$

where, $B \left( x, \delta \right)$ is the notation for the open ball around the point $x$ and of radius $\delta$ and $d \left( x, y \right)$ stands for the distance between $x$ and $y$. In $\mathbb{R}$, usually, the distance is measured by $|\cdot|$, i.e., $d\left( x, y \right) = \left| x - y \right|$.

Now, the open ball can be written as collection of all those points such that $\left| x - y \right| < \delta \Rightarrow x - \delta < y < x + \delta$. Therefore, $B \left( x, \delta \right) = \left( x - \delta, x + \delta \right)$.

Now, for your definition that you read and have posted in the link, I will first state it in words.

"Consider a point $x \in \mathbb{R}$ and a subset $E$ of $\mathbb{R}$. Then, the point $x$ will be said to be a limit point of the set $E$, iff every open ball centered at $x$ contains at least one element from $E$ other than itself".

To understand this, consider that you are in a party and suddenly a celebrity arrives at the party. Now, no matter how close you go to the celebrity, you will find at least one person from the guests of the party in that radius around the celebrity. However, if you try coming towards an ordinary person like myself, after a certain stage you will find no one close enough to me. So, we will call that celebrity a "limit point" for the members of the party while I will be yet an ordinary element from the members.

Now, coming to the explanation of definition, we shall first formulate the definition I wrote in words in Mathematical Notations using Quatifiers.

So, $x \in \mathbb{R}$ is said to be a limit point of a set $E \subseteq \mathbb{R}$ iff, $$\forall \delta > 0, \exists y \in E \text{ and } y \neq x \text{ such that } y \in B \left( x, \delta \right)$$

This in other notation can be written as the intersection of the open ball $B \left( x, \delta \right)$ excluding the point $x$ has a non - empty intersection with the set $E$, which is precisely what is written in your definition.

I hope the definition is clear! I would like you yourself to answer the other questions you have asked by using simply the definition and its understanding. Yet, if you cannot, we are there to help! But, first we would like to you to try them out.