<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] Hint: Multiply  the numerator and denominator by the conjugate of the expression.
That is, you can try working with $$\dfrac{\sqrt{(x+a)(x+b)} - x}1\cdot\frac{\sqrt{(x + a)(x + b)} + x}{\sqrt{(x + a)(x + b)} + x} = \dfrac{(x+a)(x + b) - x^2}{\sqrt{(x + a)(x+b)} + x}$$
Expanding $(x + a)(x + b)$ in the numerator and simplifying gives us 
$$ \dfrac{(a+b)x + ab}{\sqrt{x^2 + (a+b)x + ab} + x}$$
Now divide numerator and denominator by $x$ to find your limit.
$$ \lim_{x\to \infty} \dfrac{(a+b) + \frac{ab}x}{\sqrt{1 + \frac{(a+b)}x + \frac{ab}{x^2}} + 1} = \dfrac{a+b}{2}$$
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