Question

$f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}\frac{e^{\left[x\right]+\left|x\right|}}{\left[x\right]+\left|x\right|},& x\ne 0\end{array}\\ \begin{array}{cc}-1,& x=0\end{array}\end{array}$, $\left(\right[.]$ denotes the greatest integer function$),$ then

• A. $f\left(x\right)$ is continuous at $x=0$
• B. ${lim}_{x\to 0^{+}}f\left(x\right)=-1$
• C. ${lim}_{x\to 0^{-}}f\left(x\right)=1$
• D. None of these

Answer 1

The correct option is D None of these

$f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}\frac{e^{\left[x\right]+\left|x\right|}-2}{\left[x\right]+\left|x\right|},& x\ne 0\end{array}\\ \begin{array}{cc}-1,& x=0\end{array}\end{array}$

$\underset{h\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{-}}{lim}\frac{e^{\left[x\right]+\left|x\right|}-2}{\left[x\right]+\left|x\right|}=\frac{e^{-1}-2}{-1}$

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}\frac{e^{\left[x\right]+\left|x\right|}-2}{\left[x\right]+\left|x\right|}=\underset{x\to 0^{+}}{lim}\frac{e^x-2}{x}\to -\infty$