Question

If $f\left(x\right)=\{\begin{array}{cc}(1+\left|\sin x\right|{)}^{a/\left|\sin x\right|},& -\pi /6<x<0\\ b& x=0\\ e^{\tan 2x/\tan 3x}& 0<x<\pi /6\end{array}$
is a continuous function on $(-\frac{\pi }{6},\frac{\pi }{6})$, then

• A. $a=\frac{2}{3},b=e^2$
• B. $a=\frac{1}{3},b=e^{1/3}$
• C. $a=\frac{2}{3},b=e^{2/3}$
• D. none of these

Answer 1

The correct option is C $a=\frac{2}{3},b=e^{2/3}$

$b=f\left(0\right)=\underset{x\to 0^{+}}{lim}=\underset{x\to 0}{lim}{e}^{\tan 2x/\tan 3x}$
$=\underset{x\to 0}{lim}{e}^{\tan 2x/2x}\times \frac{3x}{\tan 3x}\times \frac{2}{3}=e^{2/3}$,
Also, $f\left(0\right)=\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0}{lim}(1+\left|\sin x\right|{)}^{\frac{1}{\left|\sin x\right|}\times a}=e^a$
Therefore $a=2/3$