Let $D\subseteq R$ be a subset of real numbers with an accumulation point $a.$ Let $f,g:D\to\mathbb R$ be functions with limits $\lim_{x\to a}f(x)=L$ and $\lim_{x\to a}g(x)=H.$ Then the limit of the sum of both functions is given by $$\lim_{x\to a}(f(x)+g(x))=L+H.$$
Please note that the proposition does not require the functions $f,g$ to be continuous at $a$, i.e. that $f(a)=L$ and $g(a)=H$. In this case, please refer to another proposition.
Proofs: 1