Mathematics - Continuity - Differentiability Question with Solution | TestHub

Let $S=\left\{\left(\lambda , \mu \right)\in R\times R\right\}$ and

$f\left(t\right)=\left(\left|\lambda \right|e^{\left|t\right|}-\mu \right).\mathrm{ }\sin 2\left|t\right|$

$RHD=\underset{h\to 0}{\lim }\frac{f\left(0+h\right)-0}{h}=\underset{h\to 0}{\lim }\left(\left|\lambda \right|e^h-\mu \right)\frac{\sin 2h}{h}=2(\left|\lambda \right|-\mu )$

$LHD=\underset{h\to 0}{\lim }\frac{f\left(0-h\right)-0}{-h}=\underset{h\to 0}{\lim }\left(\left|\lambda \right|e^h-\mu \right)\frac{\sin 2h}{-h}=-2(\left|\lambda \right|-\mu )$

Now, $LHD=RHD$ if $\left|\lambda \right|=\mu$

$\Rightarrow \mu \ge 0$ and $\lambda \in R$