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Question

Let f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪(1+|sinx|)a|sinx| ,π6<x<0b ,x=0etan2xtan3x ,0<x<π6.
Let a and b be such that f is continuous at x=0. Then 3(a+logb) equals

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Solution

f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪(1+|sinx|)a|sinx| ,π6<x<0b ,x=0etan2xtan3x ,0<x<π6
Since, f(x) is continuous at x=0
limh0f(0h)=f(0)=limh0f(0+h)
Now,
limh0f(0h)=limh0(1+|sin(h)|)a|sin(h)|
=limh0(1+sinh)asinh
=e(1+sinh1)asinh=ea
So,
ea=b

Also, limh0f(0+h)=limh0etan2htan3h
=limh0etan2h2h×3htan3h×23
=e2/3

ea=b=e2/3
a=23

3(a+logb)=3(23+23)=4

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