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Thread: why does this limit not exist?

  1. #1
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    why does this limit not exist?

    Why doesn't the following limit exist?
    $$\lim_{n\rightarrow \infty }\left(1+\frac{\cos(n\pi )}{n}\right)^{n}$$


    My attempt:
    Limit of $a_{2n}=(1+\frac{cos(2n\pi )}{2n})^{2n}$ is different from limit of $a_{2n+1}=(1+\frac{cos(2n\pi+\pi )}{2n+1})^{2n+1}$ because $\cos (n \pi)=(-1)^n$ which gives me $-1$ or $1$ and that's why limit doesn't exists, because cos oscilating between $-1$ and $1$
    Is my answer correct?
    Last edited by Vali; Feb 2nd 2019 at 09:00 AM.
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  2. #2
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    Re: why does this limit not exist?

    $\lim \limits_{n \to \infty} \left(1 + \dfrac{x}{n}\right)^n = e^x$

    If the limit did exist

    $\lim \limits_{n \to \infty} \left(1 + \dfrac{\cos(n\pi)}{n}\right)^n = e^{\cos(n\pi)}$

    and as you alluded to this will forever oscillate between $e$ and $e^{-1}$ so no limit exists.
    Thanks from Vali and topsquark
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  3. #3
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    Re: why does this limit not exist?

    Thank you for your help!
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