Answer
SAmple answer:
$f(x)=\displaystyle \frac{1}{x},\ g(x)=-\displaystyle \frac{1}{x}$, a=0.
Work Step by Step
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)
