Plotting $f(x) = x\lfloor 1/x \rfloor$ I want to plot $f(x) = x\lfloor 1/x \rfloor$ near the point zero for finding its limit but I can't choose proper intervals and plot it . 

I switched to $1/x$ for large positive $x,$ picture gives better idea this way. I would have preferred using a different letter, say $t = 1/x,$ but I do not yet know how to do that. So, think of the graphs as indicating that the limit as $t \rightarrow +\infty \; \; \; \lfloor t \rfloor /t = 1,$ and 
$t \rightarrow -\infty \; \; \; \lfloor t \rfloor /t = 1.$ That can then be applied to $x = 1/t$ at $0^+$ and $0^-$
Apparently the intended question is to apply the floor function again. In the graphs below, one can see easily what happens in the positive case and the negative case, when $t$ is an integer and when $t$ is not an integer. Just look at the  graphs, think of what the floor function does if applied that one more time. Note: I did try writing an extra floor function in the graph, the computer did not do it correctly. 

Next for negative $x$ 

