Answer
True
Work Step by Step
Given: $\displaystyle \lim_{x\rightarrow a}f(x)=A,\quad \displaystyle \lim_{x\rightarrow a}g(x)$ does not exist,
(*) Let us assume the opposite to the problem statement:
$\displaystyle \lim_{x\rightarrow a}[f(x)+g(x)]$ exists and equals S.
We can write $g(x)=g(x)+f(x)-f(x).$
Then,
$\displaystyle \lim_{x\rightarrow a}g(x)=\lim_{x\rightarrow a}\{[f(x)+g(x)]-f(x)\}=$
... apply Limit Law 2 (p.62)
$=\displaystyle \lim_{x\rightarrow a}[f(x)+g(x)]-\lim_{x\rightarrow a}f(x)$
$=S-A$,
which means that $\displaystyle \lim_{x\rightarrow a}g(x)$ exists.
This is in contradiction with the given terms,
so our assumption (*) was wrong.
The problem statement is true.
