Answer
False
Work Step by Step
Searching for a counterexample,
we need two functions with no limit at, say, x=0,
and we aim for their sum to be zero ( so the limit of the sum exists)
$f(x)=\displaystyle \frac{1}{x},\quad g(x)=-f(x)=-\frac{1}{x}.$
Then, neither of the limits
$\displaystyle \lim_{x\rightarrow 0}f(x),\quad \displaystyle \lim_{x\rightarrow 0}g(x)$
exist, but
$\displaystyle \lim_{x\rightarrow 0}[f(x)+g(x)]=\lim_{x\rightarrow 0}0=0$,
($\displaystyle \lim_{x\rightarrow 0}[f(x)+g(x)]$ exists)
