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Math Help - Rigorous Limits

  1. #1
    Senior Member I-Think's Avatar
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    Rigorous Limits

    We've begun limits, but the more rigorous approach I have not fully understood yet, so I'm hoping someone could help me please

    Question
    lim_{n\rightarrow{\infty}} \frac{n^2+1}{n^2+2}

    Answer
    Determine some real number L such that
    |\frac{n^2+1}{n^2+2}-L|<\epsilon

    Rough work(aka scrap work)
    BY intuitive guess, L=1. So we need
    |\frac{n^2+1}{n^2+2}-1|<\epsilon

    We need a N such that \frac{n^2+1}{n^2+2}<\epsilon+1

    Can I choose N>\frac{1}{1+\epsilon}, (by Archimedean property),
    invert to obtain \frac{1}{N}<\epsilon+1
    and proceed from there?
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  2. #2
    MHF Contributor

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    I suggest that you use \left| {\dfrac{{n^2  + 1}}{{n^2  + 2}} - 1} \right| = \left| {\dfrac{{ - 1}}{{n^2  + 2}}} \right|
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