I am supposed to prove that $\lim_{x\to \infty} [f(x)+g(x)]= L+M$.
Starting to realize I don't really understand the formal definition of a limit, although I do understand the general concept. Anyhow, so far I have:
Given $\epsilon>0$, $\exists R_f,R_g$ s.t. $|f(x)-L|<\epsilon/2$ $\forall x>R_f$ and $|g(x)-M|<\epsilon/2$ $\forall x>R_g$
Choose $R>\max\{R_f,R_g\}$. (Do we want max, min, neither?)
Suppose $x>R$. Then $x>R_f$ and $x>R_g$ whenever $|f(x)-L|<\epsilon/2$ and $|g(x)-M|<\epsilon/2$.
Then $\left\vert\big(f(x)+g(x)\big)-(L+M)\right\vert$ = $\left\vert\big(f(x)-L\big)+\big(g(x)-M\big)\right\vert$ $\le$ $|f(x)-L|+|g(x)-M|$
This implies $|f(x)-L|+|g(x)-M|$ < $\epsilon/2 + \epsilon/2$ < $\epsilon$ QED.
Am I headed in the right direction?