Question

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2-[x{]}^2,& x<2\\ a,& x=2\\ x^2-[x{]}^2+3,& x>2\end{array}$. If $f\left(x\right)$ is continuous at $x=2$, then the value of $a$ is
(where $[.]$ represents the greatest integer function and $a\in R$)

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is D $3$

Given : $f\left(x\right)=\{\begin{array}{cc}{x}^2-[x{]}^2,& x<2\\ a,& x=2\\ x^2-[x{]}^2+3,& x>2\end{array}$

Finding $L.H.L$ and $R.H.L.$

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

$\text{(OR)}$

$R.H.L.=\underset{x\to 2^{+}}{lim}{x}^2-[x{]}^2+3$
$=\underset{h\to 0}{lim}(2+h{)}^2-[2+h{]}^2+3$
$=2^2-2^2+3=3$
Since $f\left(x\right)$ is continuous at $x=0$
$\Rightarrow L.H.L.=R.H.L.=f\left(2\right)=3$
$\therefore a=3$