Let ` f(x) = {{:({1+|sin x|}^(a//|sin x|)", " pi/6 lt x lt 0),(" b, " x = 0 ),(e^(tan 2x//tan 3x) ", "0ltx ltpi/6):}`
Determine a and b such that f(x) is continous at x = 0.

Options

A. `3//2,^(3//2)`
B. `-2//3,e^(-3//2)`
C. `2//3,e^(2//3)`
D. None of these

Answer

C

Explanation

We have, `underset(x to 0^(-))limf(x)`
`=underset(c to 0)lim{1+|sinx|}^((a)/(|sin x|))`
`=e^(underset(x to 0)lim|sinx|.(a)/(|sin x|))=e^(a)`
and `underset(xto0^(-))limf(x)=underset(xto0)lime^((tan 2x)/(tan3x))`
`e^(underset(xto0)lim(tan 2x)/(2x)(3x)/(tan3x)xx2/3)`
`e^(2//3)`
For f (x) to be continous at x=0, we must have
`underset(x to 0^(-))limf(x)=underset(xto0^(+))limf(x)`
`=f(0)`
`impliese^(a)=e^(2//3)=b`
`implies a=2//3`
and ` b=e^(2//3)`