Question

Let $f\left( x \right)=\left[ x \right]+\left[ -x \right]$, where $\left[ x \right]$ denotes the greatest integer less than or equal to x.
Then, for any integer m
(a) \[\underset{x\to m}{\mathop{\lim }}\,f\left( x \right)=f\left( m \right)\]
(b) \[\underset{x\to m}{\mathop{\lim }}\,f\left( x \right)\ne f\left( m \right)\]
(c) \[\underset{x\to m}{\mathop{\lim }}\,f\left( x \right)\] does not exist
(d) None of the above

Answer 1

Hint: Try to draw the graph of $f\left( x \right)$ and analyze the graph at points of integers.

Complete step-by-step answer:

In the question we are given a function \[f\left( x \right)\] such that \[f\left( x \right)=\left[ x \right]+\left[ -x \right]\], where $\left[ x \right]$ denotes the greatest integer less than or equal to x.

Now \[f\left( x \right)=\left[ x \right]+\left[ -x \right]\] , so if x is integer then \[\left[ x \right]=x\] and \[\left[ -x \right]=-x\]

So, \[f\left( x \right)=x-x=0\]

If x is not an integer then \[\left[ x \right]=x\] and \[\left[ -x \right]=-\left[ x \right]=-1\]

So,

\[f\left( x \right)=x+\left[ -x \right]\]

\[=\left[ x \right]+\left( -\left[ x \right] \right)-1\]

\[=-1\]

So,

\[f\left( x \right)=\left\{ \begin{matrix}

   0,x\in Integer \\

   -1,x\notin Integer \\

\end{matrix} \right.\]

So, Range of \[f\left( x \right)=\left\{ 0,-1 \right\}\] and Domain of \[f\left( x \right)=R\]

Let \[m\in Integer\]

So its left hand limit is,

L.H.L \[=\underset{x\to {{m}^{-}}}{\mathop{\lim }}\,f\left( x \right)=-1\]

And it's right hand limit is,

R.H.L \[=\underset{x\to {{m}^{+}}}{\mathop{\lim }}\,f\left( x \right)=-1\]

So, \[\underset{x\to m}{\mathop{\lim }}\,f\left( x \right)\] exists.

But \[f\left( m \right)\] if m is integer \[=0\]

So, \[\underset{x\to m}{\mathop{\lim }}\,f\left( x \right)\ne f\left( m \right)\]

Hence, the correct option is ‘B’.

Note: Students should analyze the graph properly to do these types of problems easily. They should also be careful about continuity of various points.