What does "the sequence of functions has limit in R" mean?

In summary, the set T contains elements x in X such that the sequence of functions f1, f2... has a limit in R. This condition can be interpreted as both a function and a number value, as it applies to individual values of x in R and creates a function on X.
  • #1
cbarker1
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I am trying understand a condition in a set notion.
Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?
 
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  • #2
I would have to say both. The condition "f1(x),... has a limit in R" is applied for one value of ##x \in R## at a time, so it gives a single value. But it is applied for every ##x \in X##, so that gives a function on ##X##.
 
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Related to What does "the sequence of functions has limit in R" mean?

1. What is the definition of "the sequence of functions has limit in R"?

The sequence of functions has limit in R means that as the input values of a sequence of functions approach a certain value, the output values of the functions also approach a specific value in the real number system, R.

2. How do you determine if a sequence of functions has a limit in R?

To determine if a sequence of functions has a limit in R, you can use the definition of a limit, which states that for a given value, there exists a corresponding input value of the sequence of functions that produces an output value within a certain distance from the limit value. You can also use the limit laws and properties to simplify the sequence of functions and determine its limit.

3. What does it mean when a sequence of functions has a limit of infinity in R?

If a sequence of functions has a limit of infinity in R, it means that as the input values of the sequence of functions approach a certain value, the output values of the functions also approach infinity in the real number system, R. This indicates that the function is unbounded and continues to increase without bound as the input values get closer to the specified value.

4. Can a sequence of functions have more than one limit in R?

Yes, a sequence of functions can have more than one limit in R. This is known as a sequence of functions with multiple limits or a divergent sequence. In this case, the output values of the functions approach different values as the input values approach a specific value, indicating that the sequence does not converge to a single limit.

5. How is the concept of limit in R applied in real-world scenarios?

The concept of limit in R is widely used in various fields such as physics, engineering, and economics. It is used to model and predict the behavior of a system or process as it approaches a certain value. For example, in physics, the concept of limit is used to describe the behavior of a moving object as it approaches a certain speed or position. In economics, it is used to analyze the demand and supply of goods and services as they approach certain levels.

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