## Calculus 8th Edition

SAmple answer: $f(x)=\displaystyle \frac{1}{x},\ g(x)=-\displaystyle \frac{1}{x}$, a=0.
Take, for example, $f(x)=\displaystyle \frac{1}{x},\ g(x)=-\displaystyle \frac{1}{x}$, a=0. Approaching 0 from the right , $f(x)\rightarrow\infty,\ g(x)\rightarrow-\infty$ Approaching 0 from the left , $f(x)\rightarrow-\infty,\ g(x)\rightarrow\infty$ (the limits $\displaystyle \lim_{x\rightarrow 0}f(x)$ and $\displaystyle \lim_{x\rightarrow 0}g(x)$ do not exist) But, $f(x)+g(x)=0$ and $\displaystyle \lim_{x\rightarrow 0}[f(x)+g(x)] =\displaystyle \lim_{x\rightarrow 0}0=0$ (exists)