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Question

If the following function f(x) is continuous at x=0, find the values of a,b and c.
f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sin(a+1)x+sinxx,ifx<0c,ifx=0x+bx2xbx32,ifx>0.

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Solution

If f(x) is continuous at x=0, then

f(0)=f(0)=f(0+)

f(0)=limx0(a+1)sin(a+1)x(a+1)x+limx0sinxx
f(0)=a+2

f(0)=c

f(0+)=limx0x(1+bx1)bxx=limx01+bx21bx (on expanding the square root)

=12bR{0}

c=2+a=12c=12 a=122=32

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