Greatest Integer function limit problem I need help solving this problem without redefining the greatest interger function,  is that possible? 
$$\lim_{x \, \to \,\frac {1}{2}^-}[3x-\frac {1}{2}]$$
What about this rule 


Since we are interested in the behavior of $\left\lfloor 3x-\frac{1}{2}\right\rfloor$ as $x$ tends to $\frac{1}{2}$ from the left, we may assume that $\frac{1}{6}<x<\frac{1}{2}$. Then
$$\frac{3}{6}<3x<\frac{3}{2}$$
so
$$\frac{3}{6}-\frac{1}{2}<3x-\frac{1}{2}<\frac{3}{2}-\frac{1}{2}$$
which reduces to $0<3x-\frac{1}{2}<1$. It follows that $\left\lfloor 3x-\frac{1}{2}\right\rfloor=0$, so
$$\lim_{x\to\frac{1}{2}^-}\left\lfloor 3x-\frac{1}{2}\right\rfloor=0$$
