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?

For the first limit, $0<\frac3n<1$ for $n>3$, so $\left\lfloor \frac{3}{n}\right\rfloor=0$ for $n>3$. Therefore the first expression always equals zero and the limit is zero.
For the second limit, use the squeeze theorem (also called the sandwich theorem or pinch theorem) on
$$\frac{2}{n}\left( \frac{n}{3}-1\right)  \le \frac{2}{n}\left\lfloor \frac{n}{3}\right\rfloor \le \frac{2}{n}\left( \frac{n}{3}\right)$$
to get a limit of $\frac23$.
