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Math Help - Simple Limit Proof/Reasoning

  1. #1
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    Simple Limit Proof/Reasoning

    Hey all, thanks again for all help in the past! Here's a limit question that I am unsure if I am overthinking, or if it's as simple as it looks.

    It states:

    Prove:
    lim f(x) as x-->a is equal to lim f(a+h) as h-->0 (this is really just an exercise in understanding what the terms are)

    Now the easiest way would just to pop in the values and boom, lim f(a) = lim f(a). But I think he is looking for a different reasoning. My attempt was to kind of define the statements like so:

    _______

    LHS: f can be made to be as close to a limit L as desired by making x sufficiently close to a.

    RHS: f can be made to be as close to a limit L as desired by making h sufficiently close to zero.

    And in this example, it just so happens when h is made to go to zero we are left with the respective equations L's equal to each other.
    ______

    I don't think I've really "proved" anything though, have I?

    Thanks again!
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  2. #2
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    It depends on whether f(x) and f(a + h) are continuous at x = a and h = 0 respectively, and if you are allowed to use the result that \lim_{x \to n}f(x) = f(n) if f(x) is continuous at x = n.

    If so, your reasoning is fine.

    If not, you will need to use an \epsilon -\delta proof.
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