Find the constants a and b.

**fardeen_gen** · May 14th 2009, 02:37 AM

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

**Danny** · May 14th 2009, 05:04 AM

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

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

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 $b = -\frac{1}{2}.$

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