If limx→−∞ x2−x+1−ax−b=0, is

# If $\underset{x\to -\mathrm{\infty }}{lim} \left\{\sqrt{{x}^{2}-x+1}-ax-b\right\}=0$, is

1. A

$a=1,b=\frac{1}{2}$

2. B

$a=1,b=-\frac{1}{2}$

3. C

$a=-1,b=\frac{1}{2}$

4. D

none of these

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### Solution:

We have,

For and $b=1/2$ we observe that

$\underset{x\to \mathrm{\infty }}{lim} \left(\sqrt{{x}^{2}-x+1}-ax-b\right)=\underset{x\to \mathrm{\infty }}{lim} \left(\sqrt{{x}^{2}-x+1}+x-\frac{1}{2}\right)\to \mathrm{\infty }$

Hence, and $b=-\frac{1}{2}$

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