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Math Help - limit problem...

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    limit problem...

    Let b>0.

    Compute the next limit:

    lim_{n\to \infty}\frac{n}{b}[\frac{n}{b}]

    My starting:

    1-\frac{n}{b}<\frac{n}{b}[\frac{n}{b}]<1


    And I stuck here...

    (By the way I think the limit is 0)
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  2. #2
    MHF Contributor red_dog's Avatar
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    We have \left[\frac{n}{b}\right]\leq\frac{n}{b}<\left[\frac{n}{b}\right]+1

    Then \left[\frac{n}{b}\right]>\frac{n}{b}-1

    Multiply both members by \frac{n}{b}:

    \frac{n}{b}\left[\frac{n}{b}\right]>\frac{n}{b}\left(\frac{n}{b}-1\right)

    But \displaystyle\lim_{n\to\infty}\frac{n}{b}\left(\fr  ac{n}{b}-1\right)=\infty

    and applying the limit in the inequality we have that \displaystyle\lim_{n\to\infty}\frac{n}{b}\left[\frac{n}{b}\right]=\infty
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