Question about a limit definition

In summary, the conversation discusses the definition of limits in metric spaces, specifically the role of cluster points and the use of the epsilon-delta limit definition. There is confusion about the notation used, particularly with the symbols E', E", and the role of cluster points in Euclidean space. The conversation also explores the example of a simple function to better understand the concept of cluster points.
  • #1
BWV
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From Rosenlicht, Introduction to Analysis:

Definition: Let E, E′ be metric spaces, let p0 be a cluster point of E, and let f(complement(p0)) be a function. A point q ∈ E" is called a limit of f at p0 if, given any e > 0, there exists a δ > 0 such that if p ∈ E , p < > p0 and d( p, p0) < δ, then d′( f( p), q) < e.

A couple of questions, i get the standard epsilon delta limit definition as you would see it in a Calc I textbook, but the other stuff is confusing me.- what is the significance, if any, of the E’, E’’? I read E -> E’ as just showing the range and domain of the function - but what is E’’ - it seems to come out of nowhere?

- a cluster point in Euclidian space (that is all the book is concerned with) is an arbitrary ball around a limit point, so it already contains the limit point -which is outside the domain of the function- but the points around the limit point that comprise the cluster point are in the domain.

- Is q ∈ p0?

Thinking of how this would apply to a simple function like f(p) =1/p
 
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  • #2
BWV said:
From Rosenlicht, Introduction to Analysis:

Definition: Let E, E′ be metric spaces, let p0 be a cluster point of E, and let f(complement(p0)) be a function. A point q ∈ E" is called a limit of f at p0 if, given any e > 0, there exists a δ > 0 such that if p ∈ E , p < > p0 and d( p, p0) < δ, then d′( f( p), q) < e.

A couple of questions, i get the standard epsilon delta limit definition as you would see it in a Calc I textbook, but the other stuff is confusing me.- what is the significance, if any, of the E’, E’’? I read E -> E’ as just showing the range and domain of the function - but what is E’’ - it seems to come out of nowhere?

- a cluster point in Euclidian space (that is all the book is concerned with) is an arbitrary ball around a limit point, so it already contains the limit point -which is outside the domain of the function- but the points around the limit point that comprise the cluster point are in the domain.

- Is q ∈ p0?

Thinking of how this would apply to a simple function like f(p) =1/p
I can't read some of what you wrote in your Definition section
  1. let f(complement(p0)) be a function. -- In IE, I see a small dashed rectangle just to the left of f. Inside the rectangle I think it says OBJ, but I can't tell for sure.
  2. if, given any e > 0 -- The rectangle is just before e > 0.
  3. such that if p ∈ E , -- here the rectangle shows up right after E.
  4. then d′( f( p), q) < e -- The rectangle is just before e.
These don't show up at all in Google Chrome, so I don't know what the rectangles with OBJ are supposed to mean.
 
  • #3
BWV said:
- a cluster point in Euclidian space (that is all the book is concerned with) is an arbitrary ball around a limit point
No, this isn't right. A cluster point isn't a ball -- it is a single point. A ball around a point x consists of all points within the set in question that are within a certain distance from x. The difference between a limit point and a cluster point of a set X is that the limit point doesn't necessarily have to belong to X. The term "accumulation point" is also used in place of "cluster point."
 
  • #4
Ok, copied and pasted on my tablet which was probably the source of the bad characters. Thanks - ok see where I misunderstood the cluster point - that is just the point in the domain that as the function approaches the limit in the range is approached.
 
  • #5
BWV said:
Ok, copied and pasted on my tablet which was probably the source of the bad characters. Thanks - ok see where I misunderstood the cluster point - that is just the point in the domain that as the function approaches the limit in the range is approached.

A cluster point must not be part of the domain of the function.

Consider the function ##f: \mathbb{R}_0 \to \mathbb{R}: x \mapsto x/x = 1##

Then ##\lim_{x \to 0} f(x)## exists, while ##x=0## is a point that is not in the domain of the function. It is necessary though that it is a cluster point for the definition to make sense.
 
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1. What is a limit definition?

A limit definition is a mathematical concept that describes the behavior of a function as the input approaches a certain value. It is used to determine the exact value that a function approaches as the input gets closer and closer to a specific point.

2. Why is the limit definition important?

The limit definition is important because it allows us to analyze the behavior of a function and make predictions about its output. It also helps us understand the concept of continuity and allows us to solve complex problems in calculus.

3. How is a limit definition calculated?

A limit definition is calculated by finding the value of the function as the input approaches the specific point from both the left and right sides. If the values from both sides are equal, then that value is the limit of the function at that point. If the values are different, then the limit does not exist.

4. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the function as the input approaches the specific point from either the left or right side. A two-sided limit considers the behavior of the function as the input approaches the specific point from both the left and right sides simultaneously.

5. How is a limit definition used in real-life applications?

A limit definition is used in many real-life applications, such as predicting the growth of a population, analyzing the speed and acceleration of an object, and determining the maximum and minimum values of a function. It is also used in fields like engineering, physics, and economics to solve various problems and make accurate predictions.

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