Chapter 1.2, Problem 47ES

Suppose that $f$ and $g$ are two functions such that both $\underset{x\to a}{\lim }f\left(x\right)$ and $\underset{x\to a}{\lim }\left[f\left(x\right)+g\left(x\right)\right]$ exist. Use Theorem $\mathrm{1.2.2}$ to prove that $\underset{x\to a}{\lim }g\left(x\right)$ exist.