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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.

Text Solution

Verified by Experts

The correct Answer is:
`a=2/3,b=e^(2//3)`
` f(x) = {{:({1+|sin x|}^(a//|sin x|) " , "pi//6 lt x lt 0),(" , " x = 0),(e^(tan 2x//tan 3x)" , "0 lt x lt pi //6):}`
Since, f(x) is continuous at x = 0 .
`:. ` RHL (at x = 0) = LHL (at x = 0) = f(0)
` rArr underset( h to 0) lim e^(tan 2h//tan 3h) = underset( h to 0) lim { 1+ | sin h| } ^(a//| sin h|) = b`
` rArr" " e^(23) = e^(a) = b`
` :." " a = 2//3`
` and" " b = e^(2//3) `
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