# $\lim_{x\to 0} {\frac{\lfloor \frac{3}{2} +x\rfloor }{x}}$

Is the following limit exist or not? $$\lim_{x\to 0} {\frac{\lfloor \frac{3}{2} +x\rfloor }{x}}$$

I have no idea about find right-hand and left-hand limits.

• What is $[[ ]]$ Jan 25, 2017 at 20:48
• My first thought is the floor function, but that's clearly wrong... Jan 25, 2017 at 20:50
• Yes, floor function. Jan 25, 2017 at 20:51

The limit is not defined (it's infinity). Just fill in $x = 0$ and you get $\lim_{x\to0} \frac{\lfloor\frac{3}{2}+x\rfloor}{x} = \frac{\lfloor\frac{3}{2}+0\rfloor}{0} = \frac{1}{0}=\infty$. Note: for $x\to0$ with $x<0$, the limit is $-\infty$, for $x>0$ is it $+\infty$ You can also see this clearly when plotting the graph:
• Isn't $\lfloor 3/2+0\rfloor =1$? Jan 25, 2017 at 21:19