Mathematics - Continuity - Differentiability Question with Solution | TestHub

$f\left(0^{-}\right)=\underset{x\to 0^{-}}{\lim }\left(\frac{\sin ax\cos 2x}{x}+\frac{\sin 2x\cos ax}{x}+\frac{\sin x}{x}\right)$

$=\underset{x\to 0^{-}}{\lim }\left(a\cos 2x+2\cos ax+1\right)$

$=a+3$

Also, $f\left(0\right)=b$
$f\left(0^{+}\right)=\underset{x\to 0}{\lim }\frac{x^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{x^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}\left(\frac{{\left(1+3x\right)}^{\frac{1}{3}}-1}{x}\right)$, this is $\left(\frac{0}{0}\right)$form

Apply $\mathrm{LH}$ Rule 

$\Rightarrow f\left(0^{+}\right)=1$

$\therefore a=-2$

$b=1$

$\therefore a+2b=0$