Real hardwork is done when limit has reached

$\displaystyle \lim_{x \to -\infty} \left( \sqrt{x^2-x+1}-ax-b \right)=0$

$\displaystyle \lim_{x \to -\infty} \left( \sqrt{x^2-x+1}-\sqrt{a^2x^2+2abx+b^2} \right)=0$

Matching the coefficients but ignoring the constants as they're insignificant.

$1=a^2\longrightarrow a=\pm1$

$-1=2ab\longrightarrow b=\mp \frac{1}{2}$

Thus $\frac{a}{b}=\frac{\pm 1}{\mp \frac{1}{2}}=-2$

Note that the negative case is extraneous but even if you make a negative it doesn't change the answer.

Rationalize the function and replace $x$ by $-x$ . $\lim_{x \to \infty} \frac{(x^2+x+1)-(a^2x^2-2abx+b^2)}{\sqrt{x^2+x+1}-(ax-b)}$

For limit to be 0, degree of denominator must be greater than degree of numerator.

$\Rightarrow (1-a^2)=0 \text{ and } (1+2ab)=0$

We see that when $a=-1$ then $b=1/2$

And when $a=1$ then $b=-1/2$

Thus $\frac{a}{b}=-2$