Question

The value of $\underset{x\to 0}{lim}1-x+[1+x]+[1-x]$, (where $[.]$ denotes the greatest integral function) is

• A. $2$
• B. Does not exist
• C. $1$
• D. none of these

Answer 1

Answer: (A)

$2$

$\underset{x\to 0}{lim}1-x+[1+x]+[1-x]$
$LHL=\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(-h)$
$=\underset{h\to 0}{lim}1-(-h)+[1-h]+[1+h]$
$=\underset{h\to 0}{lim}1+h+0+1=2$
$RHL=\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f\left(h\right)$
$=\underset{h\to 0}{lim}1-h+[1+h]+[1-h]$
$=\underset{h\to 0}{lim}1-h+1+0=2$