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Thread: For those that never learned well the epsilon-delta proofs

  1. #46
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    Re: For those that never learned well the epsilon-delta proofs

    The \epsilon-\delta definition and proofs are the basis of all work with limits. They are the bricks from which the city of analysis is built. A solid understanding of \epsilon-\delta gives you an understanding of limits that you cannot necessarily gain simply from knowing of other results that are derived from it.
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  2. #47
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    Re: For those that never learned well the epsilon-delta proofs

    Quote Originally Posted by Archie View Post
    The \epsilon-\delta definition and proofs are the basis of all work with limits. They are the bricks from which the city of analysis is built. A solid understanding of \epsilon-\delta gives you an understanding of limits that you cannot necessarily gain simply from knowing of other results that are derived from it.
    One of the things also that I'm wondering, is if there is any situation where using/knowing epsilon-delta definition/proofs makes things easier or faster or efficient?
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  3. #48
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    Re: For those that never learned well the epsilon-delta proofs

    Generally speaking, and \epsilon-\delta proof is used to proof that the limit is some value that you already have. Other methods are usually used to find the value of the limit (if it exists). The existence of a limit often possible to show based on the continuity of the components of the function without having to resort to \epsilon-\delta. Or we might use the boundedness of a monotonic function to do the same.

    In other words, in most situations it doesn't help. That's not to say that there aren't some for which it is useful.
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  4. #49
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    Re: For those that never learned well the epsilon-delta proofs

    look epsilon numbers
    ε1>0 ,ε1<(any (r) from (R+)) so lim(ε1 +r)=r and f(x)`=lim( (f(x+ε1)-f(x))/ε1 )
    and integrity f(x+ε1)=f(x(ε2 or 0))
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