Question

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

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

Answer 1

Answer: (C)

$\frac{2}{3},e^{2/3}$

We have, $\underset{x\to 0^{-}}{lim}f\left(x\right)$

$=\underset{x\to 0}{lim}\{1+|\sin x|{\}}^{\frac{a}{|\sin x|}}$

$=e^{\underset{x\to 0}{lim}|\sin x|\cdot \frac{a}{|\sin x|}}=e^a$
and $\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0}{lim}{e}^{\frac{\tan 2x}{\tan 3x}}$

$=e^{\underset{x\to 0}{lim}}\frac{\tan 2x}{2x}\cdot \frac{3x}{\tan 3x}\times \frac{2}{3}$

$=e^{2/3}$

For $f\left(x\right)$ to be continuous at $x=0$, we must have

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

$\Rightarrow e^a=e^{2/3}=b$
$\Rightarrow a=\frac{2}{3}$ and $b=e^{2/3}$