Need for formal definition of limits

In summary, the formal definition of limits is important because it allows us to compare functions and see when they have the same limit. The life of students is made difficult because it is difficult to understand the need for the formal definition.
  • #1
Muhammad Ali
12
0
I could not understand the need for the formal definition of limits. When we know that the limit of a function 'f' exists at a point say 'a' if the the function 'f' has both right and left hand limits exist at that point 'a' and equal. Then what is the need for the formal definition, namely delta-epsilon definition? Why is the life for the students made difficult? I hope for a comprehensive answer.
 
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  • #2
Well, how would you define "right-handed" and "left-handed" limits WITHOUT the epsilon and deltas? :smile:
 
  • #3
I don't know.
 
  • #4
How would you apply "left" and "right" limits to this function:

f(x)= 0 if x is irrational, 1/n if x= m/n reduced to lowest terms

What is its limit as x approaches some number a?

In addition, the situation gets a lot more complicated with functions of more than one real variable, not to mention functions of complex variables. "Left" and "right" no longer apply.
 
  • #5
Ok, but I don't understand what we are getting by establishing a correspondence between epsilon and delta.
or
What is the purpose of constraining the difference of the function 'f(x)' and its limit say 'L' less than epsilon or similarly the difference of any x and 'a' less than delta? What is the meaning of the last sentence?
 
  • #6
Has no one drawn you a picture? Continuous, naively, means that it doesn't tear things apart. The formal interpretation of this statement is that the epsilon-delta stuff - if things start sufficiently close together (delta) then they cannot be pulled vary far apart (epsilon).

If we don't have the formal definition then how can we ever check that something is continuous? How can we teach it? It is an exact statement. It took years for people to reach the conclusion that this was the definition we wanted - it is neither God given, nor plucked from imagination.
 
  • #7
A good method for understanding definitions or proofs is to try to define or proove something in your own way, which doesn't work, so it can lead to a better understanding of the given definitions or proofs.
 

Related to Need for formal definition of limits

1. What is the purpose of having a formal definition of limits?

A formal definition of limits is essential in mathematics and science because it provides a precise and rigorous way to define and understand the concept of a limit. It allows us to make accurate calculations and predictions, and it is the foundation for many important concepts and theorems in calculus.

2. How does a formal definition of limits differ from an intuitive understanding of limits?

While an intuitive understanding of limits is based on visualizing the behavior of a function as it approaches a certain value, a formal definition of limits is based on precise mathematical language and logical reasoning. It is a more rigorous and accurate way to define and calculate limits.

3. Why is it important to have a formal definition of limits in real-world applications?

In real-world applications, such as physics, engineering, and economics, precise calculations and predictions are crucial. A formal definition of limits allows us to accurately model and analyze real-world situations and make informed decisions based on the behavior of functions.

4. How does the formal definition of limits apply to both one-sided and two-sided limits?

The formal definition of limits applies to both one-sided and two-sided limits. For a one-sided limit, the definition considers the behavior of the function on a specific side of the limit, while for a two-sided limit, the definition considers the behavior of the function on both sides of the limit.

5. Are there any limitations to the formal definition of limits?

While the formal definition of limits is a powerful tool in mathematics and science, it does have its limitations. It may not always be applicable to functions that are discontinuous or have infinite or oscillating behavior. In these cases, other methods, such as L'Hôpital's rule, may be used to evaluate limits.

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