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Simple limits question

  1. Jul 2, 2010 #1
    1. The problem statement, all variables and given/known data

    I have the function f(p,q)=p(1-p)/[q(1-q)] where p and q are in (0,1). I want to say that if p is close to q, f(p,q) is 'close' to 1. What is a formal way of saying how close to p q should be?

    3. The attempt at a solution

    Basically I want to say [tex]f(p,q)= 1 + \epsilon(p,q)[/tex] where [tex]\epsilon(p,q)[/tex] is small if some condition "X" is true. Is there an obvious way of saying what "X" is? Maybe I can substitute q=p+u... but then what? Taylor series or L'hopital's or something? My calc is pretty rusty, so I'd appreciate any reminders...
     
    Last edited: Jul 2, 2010
  2. jcsd
  3. Jul 2, 2010 #2
    Substituting p = q+u sounds like a good idea. Then use
    [tex]
    (q+u)(1-q-u) = q-q^2-2qu+u-u^2 = q(1-q)-2qu+u-u^2
    [/tex]
     
  4. Jul 2, 2010 #3
    If you define a function

    [tex]
    g(x) = x (1 - x)
    [/tex]

    can you translate the question you asked in terms of it?
     
  5. Jul 2, 2010 #4
    Why not just write that the limit[f(p,q)] = 1 as p [tex]\rightarrow[/tex] q ?
     
  6. Jul 4, 2010 #5
    I agree.
    The problem statement is nearly the very definition of a Limit.
    You just did not use the letters epsilon and delta x.
     
  7. Jul 5, 2010 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    [tex]\frac{p(1-p)}{q(1- q)}= \left(\frac{p}{q}\right)\left(\frac{1-p}{1-q}\right)[/tex]
    Does that help?
     
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