question_answer

\[Let\,\,f(x)\,\,\,=\left[ \begin{matrix}    {{\{1+|\sin x|\}}^{a/|\sin x|}}\,\,\,\,\,, & \frac{\pi }{6}<x<0  \\    b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,, & x=0  \\    {{e}^{\tan 2x/\tan 3x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}}, & 0<x<\frac{\pi }{6}  \\ \end{matrix} \right.\]. Determine a and b such that f(x) is continuous at x = 0.

A)  \[\frac{2}{3},{{e}^{2/3}}\]  

B)  \[\frac{4}{3},{{e}^{4/3}}\]

C)  \[\frac{3}{2},{{e}^{3/2}}\]                 

D)  \[\frac{1}{2},{{e}^{1/2}}\]

E)  None of these