Answer
The proof is below.
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 exist, both $\lim_{x \to a} g(x)$ and $\lim_{x \to a} f(x)$ must exist. Since $\lim_{x \to a} f(x)$ is already given to exist, $\lim_{x \to a} g(x)$ must exist.
