Math floor and limits? $\lim \limits_{n \to \infty}\frac{n}{2}\left\lfloor \frac{3}{n}\right\rfloor$ I have these equations:  
$$\lim \limits_{n \to \infty}\frac{n}{2}\left\lfloor \frac{3}{n}\right\rfloor$$    
$$\lim \limits_{n \to \infty}\frac{2}{n}\left\lfloor \frac{n}{3}\right\rfloor$$
What is the difference between the two?

If $n\gt 3$ then $\left\lfloor \dfrac3n\right\rfloor=0$ because $\dfrac3n\lt1$. Thus the first is $0$. 
The second one is equal to either $\dfrac23$, $\dfrac23\cdot\dfrac{k}{k+1}$, or $\dfrac23\cdot\dfrac{k}{k+2}$ where $n=3k+r$, with $k,r\in \mathbb Z$ integers and $r\lt3$. In the limit $n\rightarrow\infty$, $k\rightarrow\infty$, so the limit is $\dfrac23$.
