A complicated limit involving floor function

Question

Let $f(x) = \lfloor x\lfloor1/x\rfloor \rfloor $ . Find $\lim_{x \to 0^{+} } f(x) $ and $\lim_{x \to 0^{-} } f(x)$ . I think $\lim_{x \to 0^{+} } f(x)$ doesn't exist but I have no idea about $\lim_{x \to 0^{-} } f(x)$ .

Answer 1

If $1/x$ is an integer then $x\lfloor 1/x\rfloor=1$. If $1/x$ is not an integer, and $x>0$, then $x\lfloor 1/x\rfloor<1$. So the right-hand limit doesn't exist.

If $x<0$ and $1/x$ isn't an integer then $x\lfloor 1/x\rfloor>1$. However, $x\lfloor 1/x\rfloor<x(1/x-1)=1-x$, so provided $x>-1$ we have $x\lfloor 1/x\rfloor<2$.

Answer 2

Check the following graph:

You will get the answer automatically.

https://www.desmos.com/calculator/8p190y7wr2

Hint: The limit as a whole is not defined, because negative limit is not equal to positive limit.