Question: The limit I need to solve is the following. I don't know if it's correct to treat the $+2$ and the $+1$ as insignificant, or if there's another way around it.

$$ \lim\_{x \to \infty} (\sqrt{x^2+2} - \sqrt{x^2+1})$$

Step-by-step solution will be really apreciated.

Answer: 
Hint

$$\sqrt{a}-\sqrt{b} = \frac{(\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b})}{\sqrt{a}+\sqrt{b}} = \frac{a-b}{\sqrt{a}+\sqrt{b}} = \ldots$$
This has been asked many times
in many forms.

To find
$\lim \limits\_{x \to \infty} (\sqrt{x^2+a} - \sqrt{x^2+b})
$,
multiply by
$\frac{\sqrt{x^2+a} + \sqrt{x^2+b}}{\sqrt{x^2+a} + \sqrt{x^2+b}}
$
and the result is easy.