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Math Help - Binomial Theorem and Squeeze Theorem to Compute Limit

  1. #1
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    Binomial Theorem and Squeeze Theorem to Compute Limit

    Question: Use the Binomial Theorem and Squeeze Theorem to compute lim(1+(1/n^2))^n as n goes to infinity.

    I used direct substitution using the binomial theorem, but can't seem to get very far. Any ideas?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Binomial Theorem and Squeeze Theorem to Compute Limit

    I would rewrite the expression as follows:

    \left(1+\frac{1}{n^2} \right)^n=\left(\frac{n^2+1}{n^2} \right)^n=\frac{\left(n^2+1 \right)^n}{n^{2n}}

    Now, you need to find functions f(n) and g(n) such that:

    f(n)\le\left(n^2+1 \right)^n\le g(n)

    and where:

    \lim_{n\to\infty}\frac{f(n)}{n^{2n}}= \lim_{n\to\infty}\frac{g(n)}{n^{2n}}

    Can you find such functions?
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