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Math Help - What is the difference between DELTA Y and EPSILON when solving proofs?

  1. #1
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    What is the difference between DELTA Y and EPSILON when solving proofs?

    Sorry, this is probably a really stupid question. But I'd appreciate an answer

    What is the difference between DELTA Y and EPSILON when solving limit proofs?
    Last edited by mr fantastic; July 30th 2010 at 04:55 AM. Reason: Copied title into main body of post.
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  2. #2
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    So, the definition of  \mathop{\lim}\limits_{x \to c} f(x) = L is that  \forall \epsilon > 0  \exists \delta > 0 : \forall x, 0 < | x - c| < \delta \rightarrow |f(x) - L| < \epsilon.

    Whew, that's a mouthfull, so let's break that down some. Firstly, the condition is say that for any positive number  \epsilon , (no matter how super-small!), I can come up with another number  \delta > 0 so that when my x is a \delta-distance away from c,  f(x) will be within  \epsilon-distance from L.

    Think of the condition like a game. Someone gives you a number  \epsilon. All you know about it is that it's positive. You get no other information. What you're trying to do is pick a  \delta (you pick  \delta) so that the condition of the limit is satisfied.
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  3. #3
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    Just to clarify, let me work out an easy example. Say you want to compute  \mathop{\lim}\limits_{x \to 1} x = 1. Let  \epsilon > 0 be given (I don't know what it is, but it's a fixed number). Now, I need to pick a  \delta > 0 so that my limit is satisfied. But, in this case,  f(x) = x , so let  \delta = \epsilon . That is, for whatever number  \epsilon you give me, I'll pick my  \delta to be the same thing!

    So, let's test that this works. Suppose that  |x - 1| < \delta = \epsilon. Then, quite necessarily,  |f(x) - 1| = |x - 1| < \epsilon precisely because I'm assuming that  |x - c| < \delta . I hope that clears up how this process works.
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  4. #4
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    Can you give an example of a problem "solving limits" that involves a "delta y"? I have seen many proofs that a limit is as claimed that involved "epsilon" and "delta" but I have never seen one that involved "delta y".
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