$\lim\_{x\to -\infty}\sqrt{x^2+9}+x+3$

The answer is 3. I know $\sqrt{x^2+9}$ gets infinitely big ($>0$) and so does $x$ negatively, so it seems reasonable to think that the sum of them will aproach $0$ as $x$ goes to infinity, but is that true for the limit of any function $f(x) + g(x)$ when $f(x)$ goes to $\infty$ and $g(x)$ goes to $-\infty$? I tried to prove that "lemma" myself but couldn't find a way through. If that's not always true it must certainly be true for at least the situation of the presented problem, but I couldn't manage to prove that in a consistent way. Any suggestions?