Question

If $f(x)=\{\begin{array}{l}(1+|\sin x|{)}^{a/|\sin x|},-\frac{\pi }{6}<x<0\\ b,x=0\\ e^{\tan 2x/\tan 3x},0<x<-\frac{\pi }{6}\end{array},$ 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)

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