Question

If $f(x)=\{\begin{array}{ll}{\left(1+\left|sinx\right|\right)}^{\frac{a}{|sinx|}},& -\pi /6<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\pi /6\end{array}$
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 $,\underset{x\to 0^{-}}{\lim }f(x)$
$=\underset{x\to 0}{\lim }\{1+|\sin x|{\}}^{\frac{a}{|\sin x|}}$
$=e^{\underset{x\to 0}{\lim }|\sin x|\cdot \frac{a}{|\sin x|}}=e^a$
and $\underset{x\to 0^{+}}{\lim }f(x)=\underset{x\to 0}{\lim }{e}^{\frac{\tan 2x}{\tan 3x}}$
$=e^{\underset{x\to 0}{\lim }\frac{\tan 2x}{2x}\cdot \frac{3x}{\tan 3x}\times \frac{2}{3}}$
$=e^{2/3}$
For $f(x)$ to be continuous at $x=0$, we must have
$\underset{x\to 0^{-}}{\lim }f(x)=\underset{x\to 0^{+}}{\lim }f(x)$
$=f(0)$
$\Rightarrow e^a=e^{2/3}=b$
$\Rightarrow a=2/3$
and $b=e^{2/3}$