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Question

lf f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪(1+|sinx|)a|sinx|π6<x<0bx=0etan2xtan3x0<x<π6 is
continuous at x=0 then

A
a=e2/3,b=23
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B
a=23,b=e2/3
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C
a=13,b=e1/3
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D
a=e1/3,b=e1/3
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Solution

The correct option is A a=23,b=e2/3
For the function f(x) to be continuous at x=0
limx0f(x)=f(0)=limx0+f(x)
LHL
limx0exp(|sinx|×a|sinx|)=ea
f(0)=b
RHL
limx0+exp(tan2xtan3x)
=limx0+exp(tan2x2x tan3x×3x×23)=e23
Thus,
exp(a)=b=exp(23)
Hence, a=23 and b=exp(23)

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