Understanding the Difference Between Limit and Limit Point in Mathematics
Concepts of limit and limit point are fundamental in calculus and topology, but they refer to different ideas. This article aims to clarify the distinction between the two, provide examples, and discuss their significance in mathematical contexts.
What is a Limit?
A limit refers to the value that a function or sequence approaches as the input or index approaches a certain point. Formally, for a function f(x), we write limx→a f(x) L. This notation means that as the variable x gets closer to a, the function values f(x) get closer to the value L.
Examples of Limits
Consider the sequence {an} {1, 1/2, 1/3, 1/4, ...}. In this sequence, as n approaches infinity, the terms of the sequence approach zero. Therefore, we say the limit of this sequence is 0:
lim_{n→∞} a_n 0
Another example can be a function f(x) (x-1)(x-2). As x approaches 1, the value of f(x) approaches 0:
lim_{x→1} f(x) 0
What is a Limit Point?
A limit point or accumulation point of a set is a point where every neighborhood around it contains at least one point from the set different from itself. In other words, a point p is a limit point of a set S if for every ? 0, the neighborhood (p - ?, p ?) contains at least one point from S that is not p itself.
Examples of Limit Points
Consider the set S {1, 2, 3, 1, 2, 3, 1, 2, 3, ...}. The point 1 is a limit point of this set because every neighborhood around 1 contains other points of the set, such as 1’s, 2’s, and 3’s, apart from 1 itself. The same argument can be extended to points 2 and 3.
Differences Between Limit and Limit Point
The limit is associated with the behavior of functions or sequences as they approach a specific value. The limit point, however, relates to the points in a set where nearby points from the set can be found. Limits are often used in the context of functions, while limit points are used in set theory and topology.
Examples to Illustrate Differences
Example 1: Sequence {2, 0, 2, 0, 2, 0, ...}
If we look at the sequence {2, 0, 2, 0, 2, 0, ...}, both 2 and 0 are limit points because every neighborhood around these points has an infinite number of terms from the sequence. Each term an that is 2 or 0 lies within the neighborhood of the other. However, 2 and 0 are not the limits of the sequence, as the sequence does not converge to a single value.
For the sequence {2, 0, 2, 0, 2, 0, ...},lim_{n→∞} a_n is not defined, but 2 and 0 are limit points.
Example 2: Sequence {1, 1/2, 1/3, 1/4, ...}
For the sequence {1, 1/2, 1/3, 1/4, ...}, 0 is the only limit point and the limit of the sequence. As n approaches infinity, the terms of the sequence get arbitrarily close to zero. Therefore, 0 is both a limit and a limit point of the sequence, but not the only case of a limit point.
{1, 1/2, 1/3, 1/4, ...} has 0 as the limit point and limit.
Conclusion
While both limit and limit point are important concepts in mathematics, they differ in their definitions and applications. Understanding these distinctions is essential for a deeper grasp of calculus and topology. Whether dealing with functions or sets, recognizing the difference between limit and limit point can significantly enhance the clarity and accuracy of mathematical analyses.