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,
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,
If you want to actually prove this result for all cases then that would probably be a bit more trickseee (as golunm would say)