Question

If the function is $(1+|sinx|{)}^{\frac{a}{|sinx|}}$, x<0
$b,x=0$
$e^{\frac{tan2x}{tan3x}},0<x<\frac{\pi }{6}$, Continuous at x = 0, then

• A. $a=\frac{2}{3}$
• B. $b=a=e^{\frac{2}{3}}$
• C. $a=lnb$
• D. $a=\frac{3}{2}=b$

Answer 1

Answer: (A), (C)

The correct options are
A $a=\frac{2}{3}$
C $a=lnb$

$L.H.S=\underset{x\to 0}{lim}-f\left(x\right)=\underset{x\to 0}{lim}(1+sinh{)}^{\frac{a}{|sinh|}}=e^a$
$R.H.S=\underset{x\to 0}{lim}+f\left(x\right)=\underset{x\to 0}{lim}(1+sinh{)}^{\frac{a}{|sinh|}}=e^{\frac{2}{3}}$
f (x) is continuous at x = 0 $\Rightarrow$
$\underset{x\to 0}{lim}-f\left(x\right)=\underset{x\to 0}{lim}+f\left(x\right)=f\left(10\right)$
$\therefore b=e^a=e^{\frac{2}{3}}$
$\therefore a=\frac{2}{3}$.