Question

The values of $p$ and $q$ so that the function $f\left(x\right)=\{\begin{array}{cc}{|1+\sin x|}^{\frac{p}{\sin x}}& \frac{-\pi }{6}<x<0\\ q& x=0\\ e^{\frac{\sin 2x}{\sin 3x}}& 0<x<\frac{\pi }{6}\end{array}$ is continuous at $x=0$ is

• A. $p=\frac{1}{3},q=e^{2/3}$
• B. $p=0,q=e^{2/3}$
• C. $p=\frac{2}{3},q=e^{-2/3}$
• D. $p=-\frac{2}{3},q=e^{2/3}$

Answer 1

The correct option is D $p=-\frac{2}{3},q=e^{2/3}$

Given that function $f\left(x\right)$ is continuous at $x=0$

$\Rightarrow \underset{x\to 0^{-}}{lim}f\left(x\right)=f(x=0)=\underset{x\to 0^{+}}{lim}f\left(x\right)$

$f(x=0)=q$ ,
and $\underset{x\to 0^{-}}{lim}f\left(x\right)=e^{-\frac{p}{\sin x}(|1+\sin x|-1)}=e^{-p}$
and $\underset{x\to 0^{+}}{lim}f\left(x\right)=e^{\frac{2}{3}}$

$\Rightarrow e^{-p}=q=e^{\frac{2}{3}}$

$\Rightarrow p=-\frac{2}{3},q=e^{\frac{2}{3}}$

Therefore the correct option is $D$