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Math Help - Is the correct Limit?

  1. #1
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    Is the correct Limit?

    lim as n approaches infinity of (n/(n+1))^2n

    limit equals e?

    if so why?
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  2. #2
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    Quote Originally Posted by Nichelle14
    lim as n approaches infinity of (n/(n+1))^2n

    limit equals e?

    if so why?
    No,

    Consider,
    \lim_{n\to\infty}\left( 1+\frac{1}{n} \right)^n=e
    Since, the limit exists, then you can evaluate, f(x)=1/x because it is countinous for x>0.
    Thus,
    f\left( \lim_{n\to\infty}\left( 1+\frac{1}{n} \right)^n \right) =f(e)
    Thus, you have,
    \lim_{n\to\infty}\left( \frac{n}{n+1} \right)^n=1/e
    Again, since g(x)=x^2 is countinous you can evaluate both sides by it thus,
    g\left(\lim_{n\to\infty}\left( \frac{n}{n+1} \right)^n \right) =g(1/e)
    Thus,
    \lim_{n\to\infty}\left( \frac{n}{n+1} \right)^{2n}=1/e^2
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