Define:
![\[ f(x) = (-1)^{\left \lfloor \frac{1}{x} \right \rfloor} \qquad \text{for } x \neq 0, \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d238fecee3aa719581afc61b36c76251_l3.png "Rendered by QuickLaTeX.com")
where 
denotes the greatest integer function, or floor function. Sketch the graph of 
for 
and 
. Evaluate
![\[ \lim_{x \to 0^+} f(x) \qquad \text{and} \qquad \lim_{x \to 0^-} f(x). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b4b11b61972a1396261bdc11ec3ab94a_l3.png "Rendered by QuickLaTeX.com")
Is it possible to define 
in a way that makes continuous at 0.
First, we sketch the graph of 
on the requested intervals.
As 
, alternates between 
and 
.
As 
, alternates between and .
There is no way to define to make continuous at 0 since will take both values and no matter how small we choose our 
. (So, if we were to try to define 
, then for 
there is no 
such that 
whenever 
, and similarly if we try to define 
.)