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  1. #1
    la vache fantôme phantomcow2's Avatar
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    Proving limit statement --- Got delta, now what?

    I'm missing something here. I can find out what to use for delta, but how do I proceed to prove a limit statement to be correct? For example:


    let e = epsilon and d = delta

    Lim (x/5) as x -->3 = 3/5
    =|(x/5) - 3/5|
    =|(x-3)/5| Combine terms
    =.2|x-3| Factor out 1/5
    =.2|x-3| < e
    =|x-3| < 5e Divide both sides by .2

    So I know I am supposed to use 5e, right?

    Aren't I supposed to somehow prove that:
    if 0 < |x-3| < e then |(x/5)-3/5| < e

    I'm not really sure how to go about this. Can someone shed some light on the procedure?
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  2. #2
    la vache fantôme phantomcow2's Avatar
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    I feel like I am almost there but not quite. Ironically, sort of like a limit.
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  3. #3
    riding once again jschen's Avatar
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    Huh? I lost you after: Lim (x/5) as x --> 3 = 3/5

    You prove that a function reaches a limit by demonstrating that an arbitrarily small difference from the limiting value of the input (say, a deviation of d) results in an arbitrarily small change from the proposed limiting value in the function output (say, a difference of e).

    I assume you're trying to prove that Lim (x/5) as x --> 3 = 3/5. If so, how do your "equations" as they're set up do anything of that sort? Where is the arbitrarily small difference in the input value? Actually, I'm not even sure what your equation is since I don't see <terms on one side> = <terms on other side> anywhere. But since I don't see anything involving arbitrarily small values, it's not particularly promising.

    Does this shed any light on why what you've written so far doesn't make much sense conceptually?
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  4. #4
    la vache fantôme phantomcow2's Avatar
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    I'm not entirely sure how to go about proving this, though.

    What I did was what my professor called "preliminary analysis" -- evaluating an adequate delta to use to do the actual proof. The actual proof...the part I don't know how to do

    My work shown is just the profession of the algebra needed to find a delta which is used in the proof. I'm basically following my textbook's example here. I can't make sense of what my book is telling me to do past this point.
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  5. #5
    riding once again jschen's Avatar
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    Okay, let's define the parameters of the proof. To show that lim (x --> a) f(x) = y, you need to demonstrate that for any arbitrarily small deviation from the output value e (we don't know anything about e other than that it's arbitrarily small, which helps cancel out some terms), we can find a value d such that for all input values that differ from a by no more than d, the output deviates from y by no more than e.

    Reword the highly precise (but rather arcane) sentence in your mind (or on paper) in plain English. Then write the same idea in mathematical symbols. Or, just Google "limit proof" (without the quotes) and go at it without understanding the concept. If you choose the former, let me know what you come up with. If the latter, well, good luck.
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  6. #6
    riding once again jschen's Avatar
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    Oh, one more hint. Since e can be arbitrarily small, d almost certainly (though not always) will have some dependence on e. (In a trivial case, to prove that lim (x --> 3) 0 = 0, you can use whatever d you want since 0 = 0 for all values of x, so there is no need for d to have any dependence on e.) Okay, go at it. I'm not saying anything more until you come up with something, ANYTHING, that helps you out a bit (no matter how little it helps).
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  7. #7
    phony collective progress x136's Avatar
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    Do all the math you want, but the answer will still be 42.

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