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Math Help - limit identity proofs

  1. #1
    Member Greengoblin's Avatar
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    limit identity proofs

    I have these identities for limits:

    let, \lim_{x\to a} f(x)=b\text{    }\lim_{x\to a} g(x)=c

    then:

    \textcircled1 \lim_{x\to a}kf(x)=kb\text{  for a fixed constant k}

    \textcircled2 \lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x)=b\pm c

    \textcircled3 \lim_{x\to a}[f(x)g(x)]=\lim_{x\to a}f(x)\lim_{x\to a}g(x)=bc

    \textcircled4 \lim_{x\to a}\left[\frac{f(x)}{g(x)}\right]=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}=\frac{b}{c} \text{    } (c\ne 0)

    \textcircled5 \lim_{x\to a}[f(x)]^{\frac{1}{n}}=b^{\frac{1}{n}} \text{ if } [f(x)]^{\frac{1}{n}} \text{ and } b^{\frac{1}{n}} \text{ are real}

    Apparently there is some rigorous way of proving these, but my book doesn't give them. Can someone give me some hints of what I need to do? Do I use the Epsilon-delta method msomehow?

    (please don't prove, I want to have a crack myself).
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  2. #2
    Eater of Worlds
    galactus's Avatar
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    Here is a nice one on the product of limits. Yes, you can use the epsilon-delta method to show them. If you want rigor, that would be the way to go.

    http://www.math.uchicago.edu/~ershov...oduct_rule.pdf
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