Propierties of limits ad infinitum I know that $$ \lim_{x\rightarrow c}f(x) + \lim_{x\rightarrow c}g(x) = \lim_{x\rightarrow c}(f(x)+g(x))$$ holds when $c$ is finite but also is true when $c$ is $\infty$?

It gets tricky.

If $f(x) = x$ and $g(x) = -x$, then

$$\lim_{x \to \infty}f(x) + \lim_{x \to \infty}g(x) = \infty - \infty$$

And $$\lim_{x \to \infty}\left( f(x) + g(x)\right) = \lim_{x \to \infty}0 = 0$$

And recall $\infty - \infty$ is an indeterminate form so we can't just say that it is equal to $0$