Thread: Find the constants a and b.

1. Find the constants a and b.

If $\lim_{x\rightarrow -\infty} (\sqrt{x^2 - x + 1} + ax - b) = 0$, then find the constants a and b.

2. Originally Posted by fardeen_gen
If $\lim_{x\rightarrow -\infty} (\sqrt{x^2 - x + 1} + ax - b) = 0$, then find the constants a and b.
Let $x = \frac{1}{y}$ so your limit becomes

$
\lim_{y \to 0} \frac{\sqrt{y^2-y+1} + a -by}{y}
$

Performing the limit show that the limit DNE if $a \ne -1$ but you want the limit to be zero so $a = -1$. Since you how have the form $\frac{0}{0}$ you can use L'Hopita'l rule, i.e.

$\lim_{y \to 0} \frac{2y-1}{2 \sqrt{y^2-y+1}} - b$

and requiring the limit be zero gives $b = -\frac{1}{2}.$