I thought I would have found an answer here (When is the limit of a sum equal to the sum of limits?) but that question's body (and the corresponding answers) involves only a specific case and does not address, actually, the question asked in its title.
Here is my question:
Given some real-valued functions $f(x)$ and $g(x)$, is it true that if the limit $$\lim\limits_{x\rightarrow a} [f(x)+g(x)]$$ exists and has a convergent value, then $$\lim\limits_{x\rightarrow a} [f(x)+g(x)]=\lim\limits_{x\rightarrow a} f(x)+\lim\limits_{x\rightarrow a} g(x)$$ is true?
Edit 1
If the above sentence isn't universally true, would it be if we assume that both $$\lim\limits_{x\rightarrow a} f(x)\quad \text{and}\quad \lim\limits_{x\rightarrow a} g(x)$$ exist and have a convergent value?