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Piecewise functions and their limits

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Let f be the function such that f(x)=1+^2 if x is irrational and f(x)=1+4x^4 if x is rational.

    Use the Squeeze Theorem to find (lim x->0)f(x).
    Clearly, for all real numbers x, f(x)>=1. Next, we note that for a real number x,

    x^2−4x^4=(x^2)(1−4x^2)>=0
    if and only if ? >=0. This means that x^2>=4x^4 if and only if x [?,?]

    3. The attempt at a solution
    I've been stuck on this question for a while.any hints to start me off?
     
  2. jcsd
  3. Sep 20, 2012 #2

    HallsofIvy

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    If [itex]x_n[/itex] is a sequence of rational numbers, converging to 0, what is [itex]f(x_n)[/itex] for all n? What is the limit as n goes to infinity?

    If [itex]x_n[/itex] is a sequence of irrational numbers, converging to 0, what is [itex]f(x_n)[/itex] for all n? What is the limit as n goes to infinity?
     
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