## Calculus 8th Edition

Given: $\displaystyle \lim_{x\rightarrow a}f(x)=A,\quad \displaystyle \lim_{x\rightarrow a}g(x)$ does not exist, (*) Let us assume the opposite to the problem statement: $\displaystyle \lim_{x\rightarrow a}[f(x)+g(x)]$ exists and equals S. We can write $g(x)=g(x)+f(x)-f(x).$ Then, $\displaystyle \lim_{x\rightarrow a}g(x)=\lim_{x\rightarrow a}\{[f(x)+g(x)]-f(x)\}=$ ... apply Limit Law 2 (p.62) $=\displaystyle \lim_{x\rightarrow a}[f(x)+g(x)]-\lim_{x\rightarrow a}f(x)$ $=S-A$, which means that $\displaystyle \lim_{x\rightarrow a}g(x)$ exists. This is in contradiction with the given terms, so our assumption (*) was wrong. The problem statement is true.