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.
$$\lim_{x\rightarrow +\infty} \sqrt{(x-a)(x-b)}-x$$

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.
