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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.

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    There is a theorem: if both limits $\lim\limits_{a\to p}f(x)$ and $\lim\limits_{a\to p}g(x)$ exist then the limit $\lim\limits_{a\to p}(f(x) + g(x))$ exists. Could you use this one? To prove by contradiction you should suppose that the limit $\lim\limits_{a\to p}g(x)$ exists.2012-09-13
  • 0
    49k views and ONE upvote? Wow. OK, I'll make it two...2014-12-26

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