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Math Help - limits of the sum of functions at a real number

  1. #1
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    limits of the sum of functions at a real number

    Can someone give me a clue as to how to solve the following:
    Assume: \to \mathbb{R}" alt="f,g\to \mathbb{R}" />, with D\subseteq \mathbb{R} and a is an accumulation point of D.
    Prove that if \lim_{x\to a}f(x)=+\infty and \lim_{x\to a}g(x)=L, then \lim_{x\to a}(f+g)(x)=+\infty.

    also, I was wondering how to show that:
    \lim_{x\to a}(\frac{g}{f})(x)=0
    Last edited by dannyboycurtis; October 19th 2009 at 10:13 PM.
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  2. #2
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    Quote Originally Posted by dannyboycurtis View Post
    Can someone give me a clue as to how to solve the following:
    Assume: \to \mathbb{R}" alt="f,g\to \mathbb{R}" />, with D\subseteq \mathbb{R} and a is an accumulation point of D.
    Prove that if \lim_{x\to a}f(x)=+\infty and \lim_{x\to a}g(x)=L, then \lim_{x\to a}(f+g)(x)=+\infty.


    \color{red}\mbox{As}\,\,(f+g)(x)=f(x)+g(x)\,\,\mbo  x{, this follows at once from arithmetic of limits}

    also, I was wondering how to show that:
    \lim_{x\to a}(\frac{g}{f})(x)=0

    \color{red}\mbox{It's easy to show that something bounded times something going to zero} \color{red}\mbox{ must go it all to zero}

    Tonio
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  3. #3
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    Its not clear that (f+g)(x) = f(x)+g(x) because one ofthem tends to infinity, the limit laws dont hold in this case do they?
    Last edited by dannyboycurtis; October 20th 2009 at 11:33 AM.
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  4. #4
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    still not sure what to do in this case
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  5. #5
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    Quote Originally Posted by dannyboycurtis View Post
    Its not clear that (f+g)(x) = f(x)+g(x) because one ofthem tends to infinity, the limit laws dont hold in this case do they?
    Yes they do, since the other limit is finite, but you can also show it directly: is it true that

    \forall M\in\mathbb{R}\,\,\exists r\in\mathbb{R}\,\, s.t.\,\,f(x)+g(x)>M\,\,for\,\,\,x>r?

    Yes, it is true, since as \lim_{x\rightarrow\infty}g(x)=L then g(x) is bounded, so -a<g(x)<a for some
    a\in\mathbb{R} and for all x , so we need to check that f(x)+g(x)>f(x)-a , and f(x)-a can be made as large as we want as x\rightarrow\infty since \lim_{x\rightarrow\infty}f(x)=\infty!!

    Tonio
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