Analysis Help: Limit Points of a Sequence

In summary, The set A={((-1)^n)(2n/(n-4))} has two limit points, -2 and 2, due to the two convergent subsequences of the sequence (-1)^n2n/(n-4). The points in A are also isolated, and the set is open since every point in A contains a neighborhood contained in the set.
  • #1
Armbru35
7
0
Hi, I am asked to find the limit points of the given set A

A={((-1)^n)(2n/(n-4))}

so far I think 0 is a limit point, but I can't figure out if there are any more than just that one.

I believe every point in A is an isolated point, and that this set is open because every point in A contains a neighborhood contained in the set.

Any suggestions would be helpful. Thanks!
 
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  • #2
Armbru35 said:
so far I think 0 is a limit point,
Why do you think that?
Consider |An|. What happens as n goes to infinity?
I believe every point in A is an isolated point, and that this set is open
If a point is isolated, how can there be an open neighborhood?
 
  • #3
I guess I thought the limit of infinity/infinity would be 0, but now that I look at it more I feel like the limit would be 2, and because we have the (-1) in front, would the limits be -2 and 2?
 
  • #4
You are using the word "limit" very carelessly. The sequence 2n/(n-4) has limit 2. That means its "set of limit points" is {2}. The sequence [itex](-1)^n2n/(n-4)[/itex] does not converge- it has no limit. It does have two convergent subsequences- taking only even indices, it converges to 2, taking only odd indicees it converges to -2. So its "set of limits points" (not "its limits") is {-2, 2}.
 

FAQ: Analysis Help: Limit Points of a Sequence

What is a limit point?

A limit point is a point on a curve or function where the value of the function approaches a certain value as the input approaches a specific value. It is also known as a accumulation point or cluster point.

How do you determine the limit point of a function?

To determine the limit point of a function, you can use the limit definition, which states that the limit point is the value that the function approaches as the input approaches a specific value. You can also use graphical and algebraic methods to determine the limit point.

What is the significance of limit points in analysis?

Limit points are important in analysis because they help us understand the behavior of functions and curves. They also allow us to make predictions about the value of a function at specific points, and help us identify points of discontinuity or infinite limits.

Can a function have more than one limit point?

Yes, a function can have more than one limit point. In fact, a function can have an infinite number of limit points. This is because a function can approach different values as the input approaches different values.

How can limit points be used to find the continuity of a function?

Limit points can be used to determine the continuity of a function. If a function has a limit point that is equal to the value of the function at that point, then the function is continuous at that point. If there is no limit point, then the function is discontinuous at that point.

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