Let$f\left(x\right)=$ $\left\{\begin{array}{l}(1+|\sin x|{)}^{a/\left|\sin x\right|},-\frac{\pi }{6}<x<0\\ b, x=0 \\ e^{\tan 2x/\tan 3x}, 0<x<\frac{\pi }{6}\end{array}\right.$  then the values of a and b if f is continuous at $x=0$, are respectively

$f\left(x\right)=(1+|\sin x|{)}^{a/\left|\sin x\right|};-(\pi /6)<x<0=b;x=0$

$=e^{\tan 2x/\tan 3x};$ $0<x<\left(\frac{{\displaystyle \pi }}{6}\right)$ For  $f\left(x\right)$ to be continuous at $x=0$

$\underset{x\to 0-}{lim}f\left(x\right)=f\left(0\right)=\underset{x\to 0+}{lim}f\left(x\right)\underset{x\to 0-}{lim}(1+|\sin x|{)}^{a/|\sin x|}$

$=e^{\underset{x\to 0}{lim}\left(\right|\sin x|}\frac{a}{|\sin x|})=e^a.$

Now, $\underset{x\to 0+}{lim}{e}^{\tan 2x/\tan 3x}=c\underset{x\to 0+}{lim}{e}^{(\frac{\tan 2x}{2x}\times 2x)/(\frac{\tan 3x}{3x}\times 3x)}$ $=\underset{x\to 0+}{lim}{e}^{2/3}=e^{2/3}$