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Math Help - How is my understanding of N-epsilon definition of a limit?

  1. #1
    Senior Member sfspitfire23's Avatar
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    How is my understanding of N-epsilon definition of a limit?

    Now, by this definition, |\{a_n\}-L|<\epsilon. Now, epsilon is just some small number that represents how far away the sequence is from the limit. When we find a value for N we have the largest possible value for which the sequence is less than epsilon. For every n>N, n\in \mathbb{N} the sequence will be less than epsilon and so we can write -\epsilon +L<\{a_n\}<\epsilon +L.

    What would you guys add or change in my understanding? Thanks!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sfspitfire23 View Post
    Now, by this definition, |\{a_n\}-L|<\epsilon.
    Lose the brackets.

    Now, epsilon is just some small number that represents how far away the sequence is from the limit.
    Essentially, yes.

    When we find a value for N we have the largest possible value for which the sequence is less than epsilon.
    No, it is any value which makes a_n not LESS than epsilon, but makes the distance between the limit and the sequences less than epsilon. In fact, it is most often taken to be the least value for which this is true.

    For every n>N, n\in \mathbb{N} the sequence will be less than epsilon and so we can write -\epsilon +L<\{a_n\}<\epsilon +L.
    Once again it's the distance between teh limit and the sequence which is less than epsilon.
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  3. #3
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    Quote Originally Posted by sfspitfire23 View Post
    When we find a value for N we have the largest possible value for which the sequence is less than epsilon.
    Don't you mean the largest possible N for which a_n could be at 'distance' greater than or equal to  \epsilon ?

    To me, the best way to think of it is that if a_n has limit L, then for any \epsilon > 0, no matter how small, we can find N, so that if n>N, we will always have that a_n stays within distance \epsilon of L. I know that this is essentially the definition written out in words, but it helped me to get my head around it quite a bit...
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    Senior Member sfspitfire23's Avatar
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    thanks guys!
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