Question

${lim}_{x\to 1}[x-1],$ Where [.] is the greatest integer function, is equal to

• A. 1
• B. 2
• C. 0
• D. None of these

Answer 1

The correct option is D

None of these

We have, ${lim}_{x\to 1^{-}}[x-1]={lim}_{h\to 0}[1-h-1]={lim}_{h\to 0}[-h]=-1$
$(k-1<k-h<k\Rightarrow [k-h]=k-1kϵZ)$
Also,
${lim}_{x\to 1^{+}}[x-1]={lim}_{h\to 0}[1+h-1]={lim}_{h\to 0}\left[h\right]=0$
$\therefore {lim}_{x\to 1^{-}}[x-1]\ne {lim}_{x\to 1^{+}}[x-1]$