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Math Help - Proving a limit exists

  1. #1
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    Proving a limit exists

    Need a little help on the homework in the calculus book...

    I need to use an example to prove that Lim [F(x)+G(x)] as X approaches A may exist, even if neither Lim F(x) as X approaches A nor Lim G(x) as X approaches A exist.

    I know this has something to do with the "Pinch" Theorem, but I really don't know where to go with this.

    Thanks!
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  2. #2
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    Quote Originally Posted by JasonW View Post
    Need a little help on the homework in the calculus book...

    I need to use an example to prove that Lim [F(x)+G(x)] as X approaches A may exist, even if neither Lim F(x) as X approaches A nor Lim G(x) as X approaches A exist.
    Let f(x) = \left\{ \begin{array}{c}1 \mbox{ if }x\in \mathbb{Q} \\ 0 \mbox{ if }x\not \in \mathbb{Q} \end{array} \right. and g(x) = \left\{ \begin{array}{c}0\mbox{ if }x\in \mathbb{Q} \\ 1 \mbox{ if }x\not \in \mathbb{Q} \end{array} \right.

    Then for any point a\in \mathbb{R} we have that \lim_{x\to a}f(x) and \lim_{x\to a}g(x) fail to exist while \lim_{x\to a}f(x)+g(x) exists.
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  3. #3
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    I feel like I should know this already, but what does "Q" represent?
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  4. #4
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    Quote Originally Posted by JasonW View Post
    I feel like I should know this already, but what does "Q" represent?
    Rational numbers. Not "not Q" means an irrational number.
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  5. #5
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    Got it, thanks!
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