Question

For $\underset{x\to 2}{lim}{x}^2-[x{]}^2$, which of the following is/are correct?
(where [.] denotes greatest integer function)

• A. The right hand limit is $0.$
• B. The left hand limit is $3.$
• C. The left hand limit is $0.$
• D. The right hand limit is $3.$

Answer 1

Answer: (A), (B)

The left hand limit is $3.$

Given : $\underset{x\to 2}{lim}{x}^2-[x{]}^2$
Finding L.H.L and R.H.L.

$L.H.L.=\underset{x\to 2^{-}}{lim}{x}^2-[x{]}^2$
$=\underset{h\to 0}{lim}(2-h{)}^2-[2-h{]}^2$
$=2^2-1^2=3$

$R.H.L.=\underset{x\to 2^{+}}{lim}{x}^2-[x{]}^2$
$=\underset{h\to 0}{lim}(2+h{)}^2-[2+h{]}^2$
$=2^2-2^2=0$