<s>[INST] Values of the limit $\lim\limits_{x\to+\infty}\left(\sqrt{(x+a)(x+b)}-x\right)$ Hi I have a question regarding finding the values of limit for the following question.

Let $a, b \in \mathbb R$. Find the limit
$$\lim_{x\to+\infty}\left(\sqrt{(x+a)(x+b)}-x\right)$$

 [/INST] There is another way to do it and even to get a bit more information.
Let
$$y=\sqrt{(x+a) (x+b)}-x=x \left(\sqrt{1+\frac{a}{x}}\times\sqrt{1+\frac{b}{x}} -1\right)$$ and now, remembering the binomial expansion or Taylor expansion
$$\sqrt{1+\epsilon}=1+\frac{\epsilon }{2}-\frac{\epsilon ^2}{8}+O\left(\epsilon ^3\right)$$ replace $\epsilon$ successively by $\frac{a}{x}$ and $\frac{b}{x}$ to get 
$$y=x \left(\left(1+\frac{a}{2 x}-\frac{a^2}{8 x^2}+O\left(\frac{1}{x^3}\right)\right)\times\left(1+\frac{b}{2 x}-\frac{b^2}{8 x^2}+O\left(\frac{1}{x^3}\right)\right) -1\right)$$ Expand the whole stuff to get
$$y=x \left(1+\frac{a+b}{2 x}-\frac{(a-b)^2}{8 x^2}+O\left(\frac{1}{x^3}\right)-1 \right)=\frac{a+b}{2}-\frac{(a-b)^2}{8 x}+O\left(\frac{1}{x^2}\right)$$ which shows the limit and also how it is approached.
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