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Homework Help: Proof of the limit f(x) = 1/x is 1/3 near 3

  1. Feb 20, 2010 #1
    1. The problem statement, all variables and given/known data
    f(x) = 1/x
    Proof that the limit of f(x) = 1/3 near 3
    the answer provided by the text is: (1/x - 1/3) < e by requiring abs(x-3) < min(6e, 1)
    I can see that the text has the right process and therefore it reaches the right answer, but I have done it in a different process which I believe is also correct but to a different answer. Can you see where have I made the mistake in my process?

    2. Relevant equations



    3. The attempt at a solution
    This is my process:
    Note:
    (1/x - 1/3) = (3-x)/3x

    Take
    abs(x-3) < 1
    => 2 < x < 4
    => 6 < 3x < 12
    => 1/6 > 1/3x > 1/12
    => abs(1/3x) > 1/12 since 3x > 0

    abs( (3-x) / (3x) ) < e
    => abs( (3-x) ( 1/3x) ) < e

    Therefore, (1/12)(abs(3-x)) = (1/12)(abs(x-3)) < abs(3-x)abs(1/3x) < e
    shows that abs(x-3)/12 < e
    or abs(x-3) < e/ 12

    therefore, my final answer becomes
    (1/x - 1/3) < e by requiring abs(x-3) < min(e/12, 1)
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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