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Thread: What is this limit from?

  1. #1
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    What is this limit from?

    What is this limit from?-capture.png

    How is the limit = 2/3? Im not sure if l'hopital was used or how the 3^(n+1) affects the limit...
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  2. #2
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    Re: What is this limit from?

    first off realize that

    $\lim \limits_{n\to \infty} \dfrac{n^k}{3^n} = 0,~\forall k$

    next divide top and bottom by $3^n$ to obtain

    $\lim \limits_{n \to \infty}~\dfrac{2\left(1-\frac{n^3}{3^n}\right)}{3 - \frac{n^3}{3^n} - \frac{3n^2}{3^n} - \frac{3n}{3^n} - \frac{1}{3^n}}= \dfrac 2 3$
    Thanks from topsquark, HallsofIvy and Vinod
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