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Question
If f(x)= `{((sin(a+1)x+2sinx)/x,x<0),(2,x=0),((sqrt(1+bx)-1)/x,x>0):}`
is continuous at x = 0, then find the values of a and b.
Solution
Given that f is continuous at x=0
f(x)= `{((sin(a+1)x+2sinx)/x,x<0),(2,x=0),((sqrt(1+bx)-1)/x,x>0):}`
Since f x is continuous at x=0, `lim_(x->0^-)f(x)=lim_(x->0^+)f(x)=lim_(x-0)f(0)`
Thus R.H.L =`lim_(x->0^+)f(x)`
`lim_(x->0)f(0+h)`
=`lim_(h->0)(sqrt(1+bh)-1)/h`
=`lim_(h->0)(sqrt(1+bh)-1)/hxx(sqrt(1+bh)+1)/(sqrt(1+bh)+1)`
=`lim_(h->0)(1+bh-1)/(hsqrt(1+bh)+1`
=`lim_(h->0)(bh)/(hsqrt(1+bh)+1)`
=`lim_(h->0)b/(sqrt(1+bh)+1)`
=`b/2`
Given that f(0) = 2
`=>lim_(x->0^+)f(x)=f(0)`
`=>b/2=2`
⇒ b =4
Similarly,
L.H.L =`lim_(x->0^-)f(x)`
=`lim_(x->0)f(0 - h)`
=`lim_(h->0)(sin(a+1)(0-h)+2sin(0-h))/(0-h)`
=`lim_(h->0)(-sin(a+1)h-2sinh)/-h`
=`lim_(h->0)(-sin(a+1)h)/-h+lim_(h->0)(-2sinh)/-h`
=`lim_(h->0)(sin(a+1)h)/h ((a+1))/((a+1))+2lim_(h->0)sinh/h`
2 = a+1+2 `[therefore lim_(theta->0)sintheta/theta=1]`
∴ a = -1
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