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?

  • 3
    $\begingroup$ The limits of $f$ and $g$ might both fail to exist, assume for example that $g=-f$ and that $f(x)$ has no limit when $x\to a$. $\endgroup$ – Did Mar 18 '18 at 16:26
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    $\begingroup$ I agree with the commenter above. However, in the case when the limits of f and g do both exist, then the limit of f+g exists, and is the sum of the limits of f and g. $\endgroup$ – Billy Mar 18 '18 at 16:38
  • $\begingroup$ @Did If the limits of $f$ and of $g$ do not exist, can the limit of their sum exist? $\endgroup$ – Anakhand Mar 18 '18 at 16:41
  • $\begingroup$ Yes, as the example in my comment shows (did you miss it?). $\endgroup$ – Did Mar 18 '18 at 16:44
  • $\begingroup$ @Did Whoops, sorry :) — is that because if $g=-f$ then $\lim\limits_{x\rightarrow a} [f(x)+g(x)]=\lim\limits_{x\rightarrow a} [0]=0$? $\endgroup$ – Anakhand Mar 18 '18 at 16:46

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