If $\displaystyle \lim_{x\rightarrow -\infty} (\sqrt{x^2 - x + 1} + ax - b) = 0$, then find the constants a and b.
Let $\displaystyle x = \frac{1}{y}$ so your limit becomes
$\displaystyle
\lim_{y \to 0} \frac{\sqrt{y^2-y+1} + a -by}{y}
$
Performing the limit show that the limit DNE if $\displaystyle a \ne -1$ but you want the limit to be zero so $\displaystyle a = -1$. Since you how have the form $\displaystyle \frac{0}{0}$ you can use L'Hopita'l rule, i.e.
$\displaystyle \lim_{y \to 0} \frac{2y-1}{2 \sqrt{y^2-y+1}} - b $
and requiring the limit be zero gives $\displaystyle b = -\frac{1}{2}.$