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Math Help - Use definition of limit to prove lim (sqrt(1 + n^(-1))) = 1?

  1. #1
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    Use definition of limit to prove lim (sqrt(1 + n^(-1))) = 1?

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  2. #2
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    Limits

    Hello qtpipi
    Quote Originally Posted by qtpipi View Post
    We need to show that, for any \epsilon > 0, \exists N \in \mathbb{N}: \, \forall n>N,\, \sqrt{1+n^{-1}}<1+\epsilon

    So, given \epsilon >0, let \sqrt{1+x^{-1}}= 1+\epsilon

    \Rightarrow 1 + x^{-1} = 1+2\epsilon + \epsilon^2

    \Rightarrow x = \frac{1}{2\epsilon + \epsilon^2}

    So for n > \lfloor{x}\rfloor, n > \frac{1}{2\epsilon + \epsilon^2}

    \Rightarrow n^{-1} < 2\epsilon+\epsilon^2

    \Rightarrow \sqrt{1+n^{-1}} < 1 + \epsilon

    Grandad
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