Limit Problem

On rationalising : $\underset { x\rightarrow \infty }{ Lim } \frac { (\sqrt { { x }^{ 2 }-x+1 } -(ax+b))(\sqrt { { x }^{ 2 }-x+1 } +(ax+b)) }{ (\sqrt { { x }^{ 2 }-x+1 } +(ax+b)) }$

$=\underset { x\rightarrow \infty }{ Lim } \frac { { x }^{ 2 }-x+1-({ a }^{ 2 }{ x }^{ 2 }+2abx+{ b }^{ 2 }) }{ \sqrt { { x }^{ 2 }-x+1 } +(ax+b) }$

$=\underset { x\rightarrow \infty }{ Lim } \frac { (1-{ a }^{ 2 }){ x }^{ 2 }-(1+2ab)x+(1-{ b }^{ 2 }) }{ \sqrt { { x }^{ 2 }-x+1 } +(ax+b) }$

Now dividing numerator and denominator by x , we get

$=\underset { x\rightarrow \infty }{ Lim } \frac { (1-{ a }^{ 2 }){ x }-(1+2ab)+0 }{ \sqrt { 1-0+0 } +(a+0) }$ (Putting 0 where there is x in denominator as 1/x tends to zero)

Now for limit to be zero numerator should be zero and denominator should have a finite value other than zero. So $1+a\neq 0\quad \Rightarrow a\neq -1$

Now $1-{ a }^{ 2 }=0\quad \Rightarrow a=1\quad (-1\quad neglected\quad as\quad a\quad cannot\quad be\quad -1)$

Also $1+2ab=0\quad \Rightarrow 1+2b=0\quad \Rightarrow b=-1/2$

Hence answer is a=1,b=-1/2.