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Question

If f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪sin(a+2)x+sinxx,x<0b ,x=0(x+3x2)1/3x1/3x4/3,x>0 is continuous at x=0, then the value of a+2b is

A
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B
1
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C
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D
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Solution

The correct option is A 0
For f(x) to be continuous at x=0
L.H.L.=R.H.L.=f(0)=b

R.H.L.
=limx0+(x+3x2)1/3x1/3x4/3

=limh0h1/3(1+3h)1/3h1/3h1/3h

=limh0(1+3h)1/31h (00) Form
Using L- hospital's rule, we get
limh013×3×(1+3h)2/3=1
b=1

L.H.L.
=limx0(a+2)sin(a+2)x(a+2)x+limx0sinxx
=limh0(a+2)sin(a+2)h(a+2)h+limh0sinhh
L.H.L=a+2+1
a+2+1=1a=2
a+2b=0

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