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Let f(x) = {{:({1+|sin x|}^(a//|sin x|)...

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.

A

`3//2, e^(3//2)`

B

`-2//3, e^(-3//2)`

C

`2//3, e^(2//3)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
We have, `lim_(x to 0^(-))f(x)=lim_(x to 0){1+|sin x|}^((a)/(|sinx|))`
`=e^(lim_(x to 0)|sinx|*(a)/(|sinx|))=e^(a)`
`and lim_(x to 0^(+)) f(x)=lim_(x to 0) e^((tan 2x)/(tan 3x))`
`=e^(lim_(x to 0) (tan2x)/(2x)*(3x)/(tan3x)xx(2)/(3))=e^(2//3)`
For f(x) to be continuous at x = 0, we must have
`lim_(x to 0^(-))f(x)=lim_(x to 0^(+))f(x)=f(0)`
`rArr e^(a)=e^(2//3)=b`
`therefore a=2//3 and b=e^(2//3)`
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