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How do I solve? I've tried to multiply and divide by the conjugate cannot advance. $\lim_{x\rightarrow +\infty} \sqrt{(x-a)(x-b)}-x$

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Multiplying and dividing by the conjugate works fine. Let $x$ be positive and larger than $a$ and $b$. We quickly obtain $\frac{-ax-bx+ab}{\sqrt{(x-a)(x-b)}+x}.$ Divide top and bottom by $x$. (That is another commonly useful kind of move.) We get $\frac{-a-b+\frac{ab}{x}}{\sqrt{\left(1-\frac{a}{x}\right)\left(1-\frac{b}{x}\right)}+1}.$ Now finding the limit is straightforward.

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    OH, I was a bit confused on the terms, I know to mulitply by one, but was confused because I thought conjugates where just for fractions.2012-11-09
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$\bf{My\; Solution::}$ Given $\displaystyle \lim_{x\rightarrow \infty}\sqrt{(x-a)(x-b)}-x\;,$ Now Using $\bf{A.M\geq G.M}$

Now When $x\rightarrow \infty,$ Then $(x-a)\;,(x-b)\rightarrow \infty$

So $\displaystyle \frac{(x-a)+(x-b)}{2}\geq \sqrt{(x-a)(x-b)}\Rightarrow x-\left(\frac{a+b}{2}\right)\geq \sqrt{(x-a)(x-b)}$

And equality hold when $(x-a) = (x-b)\rightarrow \infty$

$\displaystyle \Rightarrow \lim_{x\rightarrow \infty}x-\left(\frac{a+b}{2}\right)-x =\lim_{x\rightarrow \infty} \sqrt{(x-a)(x-b)}-x$

So $\displaystyle \lim_{x\rightarrow \infty}\sqrt{(x-a)(x-b)}-x = -\left(\frac{a+b}{2}\right)$