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.
Best 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 $c+b$