Question: Prove that if $\lim\limits\_{x\to a} f(x)$ exists, and $\lim\limits\_{x\to a} [f(x)+g(x)]$ does not exists, then $\lim\limits\_{x\to a} g(x)$ does not exists.

I understand that I have to suppose a certain limit exists, then prove by contradication.

But which should I suppose to exists, and which should I aim towards?

(Edit)

My main question would be mainly, the logic flow of proving this question. 
Is it possible to prove 
1. directly?
2. by contrapositive?
3. by contradiction?

I believe this question is not possible to prove directly and by contrapostive, as it is impossible to show that an arbitary limit does not exist as we do not have enough infomation.

Answer: 
Hint: Assume by negation that the limit of $g(x)$ when x approaches $a$ exist and equals $b$. Denote the limit of $f(x)$ when x approaches $a$ as $c$ and prove that the limit of $f(x)+g(x)$ when x approaches is $c+b$
Just think again for a moment how proof by contradiction works.

Say, you want to prove that statement A implies statement B.

To do so, you can equivalently start with the negation of statement B and conclude that statement A is not true.

So in your case, statement B is "lim g(x) does not exist", so this should get you started with the contradiction proof.