Question

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

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

Answer 1

Answer: (C)

We have $,\displaystyle \lim _{x \rightarrow 0^{-}} f(x)$
$= \displaystyle \lim _{x \rightarrow 0}\{1+|\sin x|\}^{\frac{a}{|\sin x|}}$
$=e^{\displaystyle \lim _{x \rightarrow 0}|\sin x| \cdot \frac{a}{|\sin x|}}=e^{a}$
and $\displaystyle \lim _{x \rightarrow 0^{+}} f(x)= \displaystyle \lim _{x \rightarrow 0} e^{\frac{\tan 2 x}{\tan 3 x}}$
$=e^{\displaystyle \lim _{x \rightarrow 0} \frac{\tan 2 x}{2 x} \cdot \frac{3 x}{\tan 3 x} \times \frac{2}{3}}$
$=e^{2 / 3}$
For $f(x)$ to be continuous at $x=0$, we must have
$\displaystyle \lim _{x \rightarrow 0^{-}} f(x) = \displaystyle \lim _{x \rightarrow 0^{+}} f(x)$
$=f(0)$
$\Rightarrow e^{a} =e^{2 / 3}=b$
$\Rightarrow a=2 / 3$
and $b=e^{2 / 3}$