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Not sure how to figure out this...I tried just doing normal $\frac{(x+\frac{1}{2})} {x}$ but I know that's not solving for the floor and I know I need a squeeze theorem..but not sure how to solve this one..thanks.

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@Alex: Do you see how you've changed the potential meaning of the question? Also jtm22, if you're interested in the asymptotic behavior of this as $x\to\infty$ (as suggested in the chatroom), please divulge this information instead of leaving your readers in the dark. At any rate, if you are interested, consider dividing $x-1\le \lfloor x+1/2\rfloor\le x+1$ by $x$ and using squeeze theorem. – anon Feb 20 '13 at 20:13
@jtm22: Can you clarify whether, when you wrote $\operatorname{floor}(x+1/2)/x$, you meant $\operatorname{floor}\big((x+1/2)/x\big)$ or $\big(\operatorname{floor}(x+1/2)\big)/x$? – Rahul Feb 20 '13 at 20:38

$$ \left \lfloor \frac {x+\frac 12}x \right \rfloor = \left \lfloor 1 + \frac 1{2x} \right \rfloor = 1 + \left \lfloor \frac 1{2x} \right \rfloor = \left \{ \begin{array}{lcl} 1 & \text{if} & x \in \left (+\frac 12, +\infty \right ) \\ 0 & \text{if} & x \in \left (-\infty, -\frac 12 \right ] \\ m & \text{if} & m-1 \le \frac 1{2x} < m \\ -m & \text{if} & -m-1 \le \frac 1{2x} <-m \end{array}\right . $$ where $m \ge 1, m \in \mathbb N$

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As $x-1\leq \operatorname{floor}(x+\frac{1}{2})\leq x+1 $ So you have $$ \frac{x-1}{x}=1-\frac{1}{x}\leq \frac{\operatorname{floor}(x+\frac{1}{2})}{x} \leq \frac{x+1}{x}=1+\frac{1}{x}$$

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Note that $n=\left\lfloor\frac1{2x}\right\rfloor$ if and only if $n\le\frac1{2x}<n+1.$ Hence, $0\leq \frac1{2x}<1$ if and only if $1<2x$ if and only if $x>\frac12$, in which case $$\left\lfloor\cfrac{x+\frac12}x\right\rfloor=1+0=1.$$ Also, $-1\le\frac1{2x}<0$ if and only if $2x\le -1$ if and only if $x\le-\frac12$, in which case $$\left\lfloor\cfrac{x+\frac12}x\right\rfloor=1+-1=0,$$ so the end behavior is easy to calculate. If you want more detail, then for integers $n\neq0,-1$, we have $$n\leq\frac1{2x}<n+1$$ if and only if $$\frac1{n+1}<2x\le\frac1n$$ if and only if $$x\in\left(\frac1{2(n+1)},\frac1{2n}\right].$$ Hence, $$\left\lfloor\cfrac{x+\frac12}x\right\rfloor=\begin{cases}1 & x\in\left(\frac12,\infty\right),\\0 & x\in\left(-\infty,-\frac12\right],\\1+n & n\in\Bbb Z\smallsetminus\{-1,0\},x\in\left(\frac1{2(n+1)},\frac1{2n}\right].\end{cases}$$

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in the chat she asked for the asymptotic behaviour of floor (x+1/2) – Dominic Michaelis Feb 20 '13 at 20:19

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