Mathematics - Continuity - Differentiability Question with Solution | TestHub

$\underset{x\to 0}{\lim }f\left(x\right)=\underset{x\to 0}{\lim }\left(\frac{\ln \left(1+3x\right)}{x}-\frac{\ln \left(1-2x\right)}{x}\right)$
$=\underset{x\to 0}{\lim }\left(\frac{3\ln \left(1+3x\right)}{3x}+\frac{2\ln \left(1-2x\right)}{-2x}\right)=3+2=5$

$\left[\underset{f\left(x\right)\to 0}{\lim }\frac{\ln \left\{1+f\left(x\right)\right\}}{f\left(x\right)}=1\right]$

$\therefore f\left(x\right)$, will be continuous if $f\left(0\right)=\underset{x\to 0}{\lim }f\left(x\right)$

Thus, $k=5$