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Math Help - Question of limits of functions

  1. #1
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    Question of limits of functions

    Suppose that f and g are real valued functions defined on some interval (a,b) containing the point x_0. Suppose also that \lim_{x \to x_0}f(x) =A and \lim_{x \to x_0}f(x) =B. Prove that if f(x)<g(x) for all x \in (x_0 - \phi,x_0+\phi) (for some \phi >0). Then A \leq B. In this case is it always true that A<B.

    For the first part my proof so far consists of a contradiction. I have assumed A>B and then proceded to write out the definitions of limit tending to x_0 with \epsilon, \delta and so on. However I can't seem to find a contradiction and was wondering if someone could give me the first few lines I need.
    I also have no idea for the second part. Thanks!
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  2. #2
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    Quote Originally Posted by worc3247 View Post
    Suppose that f and g are real valued functions defined on some interval (a,b) containing the point x_0. Suppose also that \lim_{x \to x_0}f(x) =A and \lim_{x \to x_0}f(x) =B. Prove that if f(x)<g(x) for all x \in (x_0 - \phi,x_0+\phi) (for some \phi >0). Then A \leq B. In this case is it always true that A<B.
    To prove that A\le B assume that if B<A then let \epsilon=\frac{A-B}{4}}>0 .
    A contradiction quickly follows.
    Last edited by Plato; March 16th 2011 at 02:29 PM.
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  3. #3
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    Sorry, I still can't see what contradiction to get. I get that:
    0<|f(x)-A|+|g(x)-B|<\frac{A-B}{2}< and then can't see where to go from there.
    Last edited by worc3247; March 17th 2011 at 06:09 AM.
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  4. #4
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    Can you argue like this:
    0<|f(x)-A|+|g(x)-B|<\frac{A-B}{2}<\lim_{x\to\{x_0}}f(x) - \lim_{x\to\{x_0}}g(x)=\lim_{x\to\{x_0}}{(f(x)-g(x))}<0 which is a contradiction?
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  5. #5
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    Here is the idea.
    Let t \approx x_0 stand for |t-x_0|<\delta for some \delta>0.

    From the given we know that t \approx x_0 means that
    f(t) \approx A and g(t) \approx B .

    But if B<A we are given that f(t)\le g(t) there is a contradiction in there.

    Suggestion draw a picture.
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  6. #6
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    Quote Originally Posted by worc3247 View Post
    I also have no idea for the second part. Thanks!
    There might be an even simpler example, but since the limit has to be taken at an interior point of an interval I could only think of doing something like f(x)=0, g(x)=|x| if x is nonzero and g(0)=1. These are both defined on (-1,1) and g>f there. Now take limits as you approach 0.
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