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Need for formal definition of limits

  1. Nov 26, 2006 #1
    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.
  2. jcsd
  3. Nov 26, 2006 #2


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    Well, how would you define "right-handed" and "left-handed" limits WITHOUT the epsilon and deltas? :smile:
  4. Nov 26, 2006 #3
    I don't know.
  5. Nov 26, 2006 #4


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    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.
  6. Nov 26, 2006 #5
    Ok, but I don't understand what we are getting by establishing a correspondence between epsilon and delta.
    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?
  7. Nov 26, 2006 #6

    matt grime

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    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.
  8. Nov 26, 2006 #7


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    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.
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