Finding limit of $\lfloor x \rfloor + \lfloor -x \rfloor $

Question

Consider $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor $ . Now find value of $\lim_{x \to \infty} f(x) $ . I know that if $x_0 \in \mathbb{R}$ then $\lim_{x \to x_0} f(x) = -1$ but I don't know whether it is true or not in the infinity .

Answer 1

It has no limit when $x\to\infty$. Consider the sequences $$ x_n=n\qquad y_n=n+\frac{1}{2} $$ Both sequence tend to $\infty$, but notice that $f(x_n)=0$ while $f(y_n)=-1$ for every $n$.

Answer 2

Defining with $ \left\{ x \right\}$ the fractional part of $x$, i.e. $$ x = \left\lfloor x \right\rfloor + \left\{ x \right\} $$ and denoting by $[P]$ the Iverson bracket $$ \left[ P \right] = \left\{ {\begin{array}{*{20}c} 1 & {P = TRUE} \\ 0 & {P = FALSE} \\ \end{array} } \right. $$ then we have that $$ \begin{array}{l} f(x) = \left\lfloor x \right\rfloor + \left\lfloor { - x} \right\rfloor = \left\lfloor x \right\rfloor - \left\lceil x \right\rceil = - \left( {\left\lceil x \right\rceil - \left\lfloor x \right\rfloor } \right) = \\ = - \left\lceil {\left\{ x \right\}} \right\rceil = - 1 + \left[ {x \in Z} \right] = \left\{ {\begin{array}{*{20}c} { - 1} & {\left| {\;x \notin Z} \right.} \\ 0 & {\left| {\;x \in Z} \right.} \\ \end{array}} \right. \\ \end{array} $$

and the limit for $x \to \infty$ does not exist.