<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] it is easily shown that as $z$ approaches zero 
$$
\log(1+z) = z + O(z^2)
$$
so we may write
$$
\log \sqrt{(x+a)(x+b)} =\log x +  \log \sqrt{(1+\frac{a}x)(1+\frac{b}x)} \\= \log x + \frac{a+b}{2x} + O(x^{-2})
$$
taking exponentials
$$
\sqrt{(x+a)(x+b)} = x\bigg(1+O(x^{-2})\bigg) \exp\frac{a+b}{2x} \\
= (x+O(x^{-1})\bigg(1+ \frac{a+b}{2x} + O(x^{-2}\bigg) \\
= x + \frac{a+b}2 +O(x^{-1})
$$
</s>