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Math Help - How do you prove that a limit does not exist?

  1. #1
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    How do you prove that a limit does not exist?

    Hi i need help on how to prove a limit does not exist using the epsilon-delta definition of the limits.

    Its like lim x--> 5 1/x-5 does not exist

    how do i go about proving that this limit does not exist? and what do you think is a good approach to proving limits using the definition?

    Thank you,
    Angelo
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by gello88 View Post
    Hi i need help on how to prove a limit does not exist using the epsilon-delta definition of the limits.

    Its like \lim_{x\to5}\frac{1}{x-5} does not exist

    how do i go about proving that this limit does not exist? and what do you think is a good approach to proving limits using the definition?

    Thank you,
    Angelo
    This question was asked and answered earlier here.
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Yeah, seriously.
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    Quote Originally Posted by gello88 View Post
    Hi i need help on how to prove a limit does not exist using the epsilon-delta definition of the limits.

    Its like lim x--> 5 1/x-5 does not exist

    how do i go about proving that this limit does not exist? and what do you think is a good approach to proving limits using the definition?

    Thank you,
    Angelo
    I'm not sure but I think you meant \lim_{x\to 5}\frac{1}{x-5}...so suppose the limit exists and equals L, then:

    \forall \epsilon >0\,\,\exists N_\epsilon \in \mathbb{N}\,\,s.t.\,\,\forall\,\,n>N_\epsilon\,,\, \,\mid \frac{1}{x-5}-L\mid <\epsilon \Longleftrightarrow L-\epsilon <\frac{1}{x-5}< L+\epsilon (*)

    On the other hand, the expression \frac{1}{x-5} can be made as big (as negative big) as wanted by choosing x close enough to 5 from the right (from the left): \frac{1}{x-5}>K\longrightarrow x-5<\frac{1}{K}, and this contradicts (*) above since there \frac{1}{x-5} is bounded...

    - If L=\infty\,(-\infty) the proof is easier, by making x approach 5 from the left (from the right).

    - Tonio
    Last edited by mr fantastic; November 5th 2009 at 04:02 AM. Reason: Restored a nice reply (no need to throw it out!)
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  5. #5
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    ouu thats right, sorry about that.

    But the answer there was handled using one sided limits which I know is a very good way of proving it and it works. But my question is how do i go about proving this limit does not exist using the:

    epsilon - delta definition?

    ∀ε > 0, ∃δ > 0 such that 0<|x-c|<δ ---> |f(x) - L| < ε
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