# What is the difference between $\lim_{x \to 0^{-}}\lfloor x \rfloor$ and $\lfloor\lim_{x \to 0^{-}} x\rfloor$

According to Wolfram Alpha : $\lim_{x \to 0^{-}}\lfloor x \rfloor = -1$ and $\lfloor\lim_{x \to 0^{-}} x\rfloor = 0$ . The first expression is obvious but the second doesn't make sense . It should be $-1$ because for example we have $\lfloor - 0.00001 \rfloor = -1$ . My teacher also accepted the Wolfram Alpha's result . I'm really confused about it .

Hint: $\lim_{x\rightarrow 0^{-}}{x}=0$.
This is due the fact that the function $$x \mapsto \lfloor x\rfloor$$ isn't conutinuous. You have to be aware of how a limit works. For the first term $$\lim_{x\to 0-} \lfloor x\rfloor$$ you take values which have a small distance to $$0$$ and are negative. In other words for every $$\epsilon>0$$ you take $$y \in (-\epsilon,0)$$ and for all these $$y$$ you have $$\lfloor y \rfloor = -1.$$ Hence, no matter how close you come to $$0$$ the value of the bracket will always be $$-1$$, which means nothing else than the limit is also $$-1$$.
On the other hand we have $$\lim_{x\to 0 -} x= 0$$ which is pretty obvious. Now if you insert that into the floor brackets, you obatin $$\lfloor\lim_{x\to 0 -} x\rfloor = \lfloor0\rfloor=0$$