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Finding the limit without L'Hôpital's rule

  1. Jan 10, 2013 #1
    1. The problem statement, all variables and given/known data

    Required to prove that
    \displaystyle\lim_{n\rightarrow \infty} ((1 - \frac{1}{n^2})^{n}) = 1

    2. Relevant equations

    [itex]\displaystyle\lim_{n\rightarrow \infty} ((1 + \frac{1}{n})^{n})[/itex] is bounded above by e. I'm not sure if this is relevant, but it was the first part of the question, so I'd assume so?

    Also, we haven't proved L'Hopital's rule yet, so I can't use that.

    3. The attempt at a solution

    I was thinking to maybe try and write it in a similar way to the first part.

    So: [itex]
    \displaystyle\lim_{n\rightarrow \infty} (((1 + \frac{1}{-(n^2)})^{-(n^2)})^{\frac{-1}{n}})

    But, as n tends to infinity [itex]-n^2[/itex] tends to negative infinity?
  2. jcsd
  3. Jan 10, 2013 #2


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    Homework Helper

    Try the generalised binomial theorem. If you want to prove rigorously that all the terms except the first go to zero at the limit, use the squeeze theorem.
  4. Jan 13, 2013 #3
    yes, it is relevant...firstly, u should understand why [itex]\displaystyle\lim_{n\rightarrow \infty} ((1 + \frac{1}{n})^{n}) = e [/itex] then u can solve this question easily..
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