How do I calculate $\lim_{x\to+\infty}\sqrt{x+a}-\sqrt{x}$? I've seen a handful of exercises like this:
$$\lim_{x\to+\infty}(\sqrt{x+a}-\sqrt{x})$$
I've never worked with limits to infinity when there is some arbitrary number $a$. I am not given any details about it.
Apparently the answer is $0$. How was that conclusion reached?

My guess is that since $x = +\infty$, the result of $x + a$ will still be $+\infty$ so we would have $\sqrt{x}-\sqrt{x} = 0$.
But that doesn't convince me. For starters,  we don't know what $a$ is: it could be $-\infty$ or something, so $\infty - \infty$ would be indeterminate...

You may write
$$
\sqrt{x+a}-\sqrt{x}=\left(\sqrt{x+a}-\sqrt{x}\right)\frac{\sqrt{x+a} + \sqrt{x}}{\sqrt{x+a} + \sqrt{x}}=\frac{a}{\sqrt{x+a} + \sqrt{x}}
$$ then it becomes easier to obtain your limit as $x \to \infty$.
