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Math Help - Find the constants a and b.

  1. #1
    Super Member fardeen_gen's Avatar
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    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.
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  2. #2
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    Quote Originally Posted by fardeen_gen View Post
    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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