Solutions

Suppose $\lim_{x\to 0}g(x)$ does not exist. In the forward direction, suppose that

$$\lim_{x\to 0}[f(x)+g(x)]$$

does not exist for some function $f$.

Now, for contradiction, assume that

$$\lim_{x\to 0}f(x)$$

does not exist. Choose $g(x)=-f(x)$. It follows that

$$\lim_{x\to 0}[f(x)+g(x)]=\lim_{x\to 0}0=0,$$

but we originally claimed that the limit does not exist.

To prove the converse $(\Leftarrow)$, suppose

$$\lim_{x\to 0}f(x)$$

exists. By Exercise 8(a),

$$\lim_{x\to 0}[f(x)+g(x)]$$

does not exist.