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Thread: limit problem...

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

    Let $\displaystyle b>0$.

    Compute the next limit:

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

    My starting:

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


    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 $\displaystyle \left[\frac{n}{b}\right]\leq\frac{n}{b}<\left[\frac{n}{b}\right]+1$

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

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

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

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

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