I have the following problem. Does this limit exist:
$$\lim_{x\rightarrow 0}|x|\cdot(-1)^{\lfloor1/x\rfloor}$$
where $\lfloor\cdot\rfloor$ is a integer part function (greatest integer not greater than given number).
Does the squeeze theorem hold in this case? Could I say that since $$-|x|\le|x|\cdot(-1)^{\lfloor1/x\rfloor}\le|x|$$ and $\lim_{x\rightarrow 0}|x|=0$ the limit under consideration exists and equals to 0?
Yes, the squeeze theorem applies and you have used it correctly to show that $$ \lim_{x\rightarrow 0}|x|\cdot(-1)^{\lfloor1/x\rfloor} = 0 $$.