Can Calculus/Analysis Help Determine Point Limits?

In summary, the conversation discusses the concept of a limit in calculus/analysis and clarifies the difference between a single point and a sequence of points. A point is said to be a limit of a sequence if every open neighborhood of that point contains all but a finite number of the points in the sequence. However, there may be different interpretations and definitions of a limit depending on the context.
  • #1
Blue and green
16
0
Can someone rephrase the title question into something more meaningful in terms of Calculus/Analysis?
 
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  • #2
Possibly kindly answer it as well. Thanks.
- blue
 
  • #3
A single point does not but a series of points can approach a limit.
 
  • #4
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
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
Fredrik said:
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!
 
  • #7
Note that sequences may have many limits. If your topological space is not hausdorff, this may happen.

Fredrik said:
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.

pwsnafu said:
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,...)
 
  • #8
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
 

Related to Can Calculus/Analysis Help Determine Point Limits?

1. What is a point limit in calculus?

A point limit in calculus refers to the value that a function approaches as a specific variable approaches a given point. It is an important concept in calculus and is used to determine the behavior of a function near a specific point.

2. How does calculus help determine point limits?

Calculus provides a set of tools and techniques to analyze the behavior of a function at a specific point. By using the concepts of continuity, derivatives, and limits, calculus can help determine the value of a point limit and understand the behavior of a function near that point.

3. Can calculus determine point limits for all functions?

No, calculus cannot determine point limits for all functions. There are some functions that are not continuous or differentiable at a given point, making it impossible to calculate the point limit using calculus. In these cases, other mathematical methods may be used to determine the point limit.

4. How is the concept of infinity related to point limits in calculus?

In calculus, infinity is often used as a way to describe the behavior of a function at a specific point. A point limit can approach infinity or negative infinity, indicating that the function is increasing or decreasing without bound near that point. Calculus helps to determine these infinite limits and understand the behavior of the function at those points.

5. Can calculus be used to determine point limits in real-world applications?

Yes, calculus is commonly used in real-world applications to determine point limits. For example, it can be used in physics to calculate the velocity or acceleration of an object at a specific point in time, or in economics to determine the maximum profit or loss of a business at a certain point in production. Calculus is a powerful tool for understanding and analyzing the behavior of functions in real-world scenarios.

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