Limits of functions and sequences

In summary, the conversation discusses the concept of functions and limits in different mathematical spaces, including euclidean spaces and topological spaces. The topic of continuity and differentiation is brought up, as well as the possibility of a function having limits at points. The existence and uniqueness of limits in different types of spaces, such as metric spaces and Hausdorff spaces, is also discussed. The conversation ends with a mention of the Zariski topology and its implications on the concept of limits in algebraic geometry.
  • #36
excellent point! you are quite right that one should consider also possibly reducible curves, but i tend to think of irreducible ones. So, good point well taken, a closed subset of a reducible curve may not be finite, it might be a whole component! so maybe add "irreducible" to my statements about curves.
 
  • Like
Likes S.G. Janssens
Physics news on Phys.org
  • #37
mathwonk said:
but i tend to think of irreducible ones.
I think this makes a lot of sense, given your background.

Thank you, and also @WWGD, for the little discussion.
 
<H2>1. What is the definition of a limit in mathematics?</H2><p>A limit is a fundamental concept in calculus that describes the behavior of a function or sequence as its input or index approaches a certain value. It is denoted by the symbol "lim" and is often used to describe the behavior of a function at a specific point or as its input approaches infinity or negative infinity.</p><H2>2. How do you determine the limit of a function?</H2><p>The limit of a function can be determined by evaluating the function at values that are very close to the desired input value. This can be done by plugging in values on either side of the desired input value and observing the output. If the outputs approach a specific value as the inputs get closer to the desired value, then that value is the limit of the function.</p><H2>3. What is the difference between a one-sided limit and a two-sided limit?</H2><p>A one-sided limit only considers the behavior of a function as the input approaches from one side, either the left or the right. A two-sided limit, on the other hand, takes into account the behavior of the function as the input approaches from both the left and the right. In order for a two-sided limit to exist, the one-sided limits must be equal.</p><H2>4. Can a function have a limit at a point where it is not defined?</H2><p>Yes, a function can have a limit at a point where it is not defined. This is known as a removable discontinuity. In this case, the limit is equal to the value that the function could be made to approach by redefining the function at that point.</p><H2>5. What is the connection between limits and continuity?</H2><p>Limits and continuity are closely related concepts. A function is continuous at a point if its limit at that point exists and is equal to the function's value at that point. In other words, a function is continuous if it has no breaks or jumps in its graph. Limits can also be used to determine if a function is continuous at a given point.</p>

1. What is the definition of a limit in mathematics?

A limit is a fundamental concept in calculus that describes the behavior of a function or sequence as its input or index approaches a certain value. It is denoted by the symbol "lim" and is often used to describe the behavior of a function at a specific point or as its input approaches infinity or negative infinity.

2. How do you determine the limit of a function?

The limit of a function can be determined by evaluating the function at values that are very close to the desired input value. This can be done by plugging in values on either side of the desired input value and observing the output. If the outputs approach a specific value as the inputs get closer to the desired value, then that value is the limit of the function.

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

A one-sided limit only considers the behavior of a function as the input approaches from one side, either the left or the right. A two-sided limit, on the other hand, takes into account the behavior of the function as the input approaches from both the left and the right. In order for a two-sided limit to exist, the one-sided limits must be equal.

4. Can a function have a limit at a point where it is not defined?

Yes, a function can have a limit at a point where it is not defined. This is known as a removable discontinuity. In this case, the limit is equal to the value that the function could be made to approach by redefining the function at that point.

5. What is the connection between limits and continuity?

Limits and continuity are closely related concepts. A function is continuous at a point if its limit at that point exists and is equal to the function's value at that point. In other words, a function is continuous if it has no breaks or jumps in its graph. Limits can also be used to determine if a function is continuous at a given point.

Similar threads

Replies
1
Views
820
Replies
3
Views
975
Replies
11
Views
1K
  • Calculus
Replies
7
Views
2K
  • Calculus
Replies
4
Views
1K
Replies
2
Views
168
Replies
11
Views
1K
Replies
2
Views
689
Replies
7
Views
841
  • Science and Math Textbooks
Replies
3
Views
691
Back
Top