Chapter 1.2, Problem 46ES

Suppose that $f$ and $g$ are two functions such that ${\lim }_{x\to a}f\left(x\right)$ exists but ${\lim }_{x\to a}\left[f\left(x\right)+g\left(x\right)\right]$ does not exist. Use Theorem $\mathrm{1.2.2}$ to prove that ${\lim }_{x\to a}g\left(x\right)$ does not exist.