Answer
Does not exist
Work Step by Step
By Theorem 1.2.2., $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x)
+ \lim_{x \to a} g(x) = R + \lim_{x \to a} g(x) $
where $R$ is a real number.
Therefore, for $\lim_{x \to a} [f(x) + g(x)]$ to not exist, $\lim_{x \to a} g(x) $ must not exist as we have already reached a real value from $\lim_{x \to a} f(x)$.
