Question

If the function
$f\left(x\right)=\{\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|,}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

• A. $a=lo{g}_{e}b,a=\frac{2}{3}$
• B. $b=lo{g}_{e}a,a=\frac{2}{3}$
• C. $a=lo{g}_{e}b,b=2$
• D. none of these

Answer 1

The correct option is A

$a=lo{g}_{e}b,a=\frac{2}{3}$

$LHL={lim}_{x\to 0^{-}}f\left(x\right)={lim}_{h\to 0}f(0-h)={lim}_{h\to 0}(1+(sin(-h)){)}^{\frac{a}{|sin(-h)|}}Let|sinx|=k.Asx\to 0,k\to 0LHL={lim}_{k\to 0}(1+k{)}^{\frac{a}{k}}=e^a$

$RHL={lim}_{x\to 0^{+}}f\left(x\right)={lim}_{h\to 0}f\left(h\right)={lim}_{h\to 0}{e}^{\frac{tan2h}{tan3h}}=e^{{lim}_{h\to 0}\frac{2}{3}.\frac{tan2h}{2h}.\frac{3h}{tan3h}}=e^{\frac{2}{3}}$

Since f(x) is continuous at x = 0,
$e^a=b=e^{\frac{2}{3}}\Rightarrow a=\frac{2}{3},a=lo{g}_{e}b$