# How to solve $\lim_{x\rightarrow +\infty} \sqrt{(x-a)(x-b)}-x$?

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.
I was answering the OP, who knew about the conjugate. Actually, it is not quite the conjugate, but I went along with it. Multiply top and bottom (which is $1$) by $\sqrt{(x-a)(x-b)}+x$. On top we get $(x-a)(x-b)-x^2$, which simplifies to $-ax-bx+ab$. Multiplying by a conjugate is a widely useful trick. When a problem involves $a+b\sqrt{d}$, it is often useful to get $a-b\sqrt{d}$ involved. You may have seen this with complex numbers. There, when you see $a+bi$, the number $a-bi$ is often helpful. –  André Nicolas Nov 9 '12 at 6:54