How could I show by means of an example that lim x ->a [f(x) + g(x)] may exist even though neither lim x->a f(x) nor lim x->a g(x) exists?
If it is an example you seek,
$\displaystyle \mathop {\lim} \limits_{ x \to 3} ( \frac{1}{x-3} - \frac{6}{x^2-9}) $
Where a = 3 and f(x) is the left function and g(x) is the right function. We can see that neither of these limits exist by themselves! But, we can combine them,
$\displaystyle \mathop {\lim} \limits_{ x \to 3} ( \frac{1}{x-3} - \frac{6}{x^2-9}) = \mathop {\lim} \limits_{ x \to 3} ( \frac{1}{x-3} - \frac{6}{(x-3)(x+3)}) = \mathop {\lim} \limits_{ x \to 3} \frac{ x+3 - 6}{(x-3)(x+3)} = \mathop {\lim} \limits_{ x \to 3} \frac{ x- 3}{(x-3)(x+3)} = \frac{1}{6} $
If you want to actually prove this result for all cases then that would probably be a bit more trickseee (as golunm would say)