# lim_(x->-oo)(x*|__(1/x)__|)?

Jun 17, 2018

The limit does not exist (diverges to $+ \infty$)

#### Explanation:

For $x < - 1$, we have

$0 > \frac{1}{x} > - 1$

and hence

$\lfloor \left(\frac{1}{x}\right) \rfloor = - 1$

So, for $x < - 1$, we have

$x \setminus \lfloor \frac{1}{x} \rfloor = - x$

and the limit we are seeking is

${\lim}_{x \to - \infty} x \setminus \lfloor \frac{1}{x} \rfloor = {\lim}_{x \to - \infty} \left(- x\right)$

It is clear that this limit does not exist (diverges to $+ \infty$)