## Calculus 8th Edition

Searching for a counterexample, we need two functions with no limit at, say, x=0, and we aim for their sum to be zero ( so the limit of the sum exists) $f(x)=\displaystyle \frac{1}{x},\quad g(x)=-f(x)=-\frac{1}{x}.$ Then, neither of the limits $\displaystyle \lim_{x\rightarrow 0}f(x),\quad \displaystyle \lim_{x\rightarrow 0}g(x)$ exist, but $\displaystyle \lim_{x\rightarrow 0}[f(x)+g(x)]=\lim_{x\rightarrow 0}0=0$, ($\displaystyle \lim_{x\rightarrow 0}[f(x)+g(x)]$ exists)