Question

If f : R $\to$ R is defined by $f\left(x\right)=[x-3]+|x-4|$ ( $\left[x\right]$ denotes greatest integer function) for $x\in R$, then $\underset{x\to 3^{-}}{lim}f\left(x\right)$ is equal to

• A. $-2$
• B. $-1$
• C. $0$
• D. $1$

Answer 1

The correct option is C $0$

We have,
$f\left(x\right)=[x-3]+|x-4|$

$\therefore \underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{-}}{lim}[x-3]+\underset{x\to 3^{-}}{lim}|x-4|$

$\Rightarrow \underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}[3-h-3]+\underset{h\to 0}{lim}|3-h-4|$

$\Rightarrow {lim}_{x\to 3^{-}}f\left(x\right)={lim}_{h\to 0}[-h]+{lim}_{h\to 0}|-1-h|$

$\Rightarrow \underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}-1+\underset{h\to 0}{lim}(1+h)=-1+1=0$