limit problem...

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July 9th 2010, 09:44 PM
Also sprach Zarathustra

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)
July 9th 2010, 11:46 PM
red_dog

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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