Question

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

• A. $\frac{2}{3}, \frac{3}{2}$
• B. $\frac{2}{3}, e^{2/3}$
• C. $\frac{3}{2}, e^{3/2}$
• D. None of these

Answer 1

Answer: (B)

$\begin{cases}
f(x)=(1+|\sin x|)^{a /|\sin x|}, & -\frac{\pi}{6} < x < 0 \\
b & , x=0 \\
=e^{\tan 2 x / \tan 3 x} & , 0 < x < \frac{\pi}{6}
\end{cases}$
For $f(x)$ to be continuous at $x=0$
Now, $\displaystyle\lim _{0} e^{\tan 2 x / \tan 3 x}$
$\displaystyle\lim _{x \rightarrow 0^{+}}$
$=\displaystyle\lim _{x \rightarrow 0^{+}} e^{\left(\frac{\tan 2 x}{2 x} \times 2 x\right) /\left(\frac{\tan 3 x}{3 x} \times 3 x\right)} $
$=\displaystyle\lim _{x \rightarrow 0^{+}} e^{2 / 3}=e^{2 / 3}$
Since, $f(x)$ is continuous at $x=0$.
$\therefore e^{a}=e^{2 / 3} \Rightarrow a=\frac{2}{3}$
and $b=e^{2 / 3}$