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Proof of theorem (Limit)

  1. Oct 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that if f(x)<=g(x) then lim f(x) <= lim g(x).


    2. Relevant equations
    -


    3. The attempt at a solution

    I've tried by definition of limit, but I didn't get anywhere with this... Can anyone help me??
     
  2. jcsd
  3. Oct 5, 2013 #2

    phinds

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    Does the limit of a function ever give you a value that is higher than the range of values of the function itself?
     
  4. Oct 5, 2013 #3

    HallsofIvy

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    Try a "proof by contradiction". Suppose lim f(x)> lim g(x). Let [itex]\alpha[/itex]= lim f(x)- lim g(x) and choose [itex]\epsilon= \alpha/2[/itex].

    I am puzzled by phind's question. The answer is "yes, it does" but I don't see how that helps here.

    (Note, by the way, if the condition were "f(x)< g(x)" then it would NOT be true that "lim f(x)< lim g(x)". Phind's suggestion would be helpful in proving that.)
     
  5. Oct 5, 2013 #4

    phinds

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  6. Oct 5, 2013 #5

    LCKurtz

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    Try ##\lim_{x\to\infty}\frac 1 x##.
     
    Last edited: Oct 5, 2013
  7. Oct 5, 2013 #6

    phinds

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    OK, thanks. I got it.
     
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