Finding limit of $\lfloor x \rfloor + \lfloor -x \rfloor $ 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 . 
 A: 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 $n$.
A: 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.
