Question

If $f\left(x\right)=\{\begin{array}{cc}x,& x<0\\ 1,& x=0\\ x^2,& x>0\end{array},$ then $\underset{x\to 0}{lim}f\left(x\right)$ is equal to:

• A. $0$
• B. $1$
• C. $2$
• D. does not exist

Answer 1

The correct option is A $0$

$f\left(x\right)=\{\begin{array}{cc}x,& x<0\\ 1,& x=0\\ x^2,& x>0\end{array}$

$LHL=\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}x=0$
$(\because x\to 0^{-}$ implies that $x<0$ for which $f\left(x\right)=x)$

$RHL=\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}{x}^2=0$
$(\because x\to 0^{+}$ implies that $x>0$ for which $f\left(x\right)=x^2)$

Clearly, $LHL=RHL$
So, $\underset{x\to 0}{lim}f\left(x\right)=0$