Question

$Letf\left(x\right)=\{\begin{array}{cc}{(1+|sinx|)}^{\frac{a}{|sinx|}},& \frac{-x}{6}<x<0\\ b,& x=0\\ e^{tan2x/tan3x},& 0<x<\frac{-\pi }{6}\end{array}$ then the value of a and b if f is continuous at x=0, are respectively

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

Answer 1

The correct option is B $\frac{2}{3},e^{\frac{2}{3}}$

$f\left(x\right)=\{\begin{array}{cc}{(1+|sinx|)}^{\frac{a}{|sinx|}},& \frac{-x}{6}<x<0\\ b,& x=0\\ e^{tan2x/tan3x},& 0<x<\frac{-\pi }{6}\end{array}$
For f(x) to be continuous at x = 0

$\underset{x\to 0-}{lim}f\left(x\right)=f\left(0\right)=\underset{x\to 0+}{lim}f\left(x\right)$

$\underset{x\to 0-}{lim}(1+|sinx|{)}^{\frac{a}{|sinx|}}=e^{\underset{x\to 0}{lim}\left(|sinx|\frac{a}{|sinx|}\right)}=e^a.$

$Now,\underset{x\to 0+}{lim}{e}^{\frac{tan2x}{tan3x}}=c\underset{x\to 0+}{lim}.e^{(\frac{tan2x}{2x}\times 2x)/(\frac{tan3x}{3x}\times 3x)}$

$=\underset{x\to 0+}{lim}{e}^{\frac{2}{3}}=e^{\frac{2}{3}}$