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Problem proving if a limit exists

  1. Feb 26, 2012 #1
    1. The problem statement, all variables and given/known data
    We are given two functions f : [itex]\mathbb{R}^n[/itex] -> [itex]\mathbb{R}[/itex] and g: [itex]\mathbb{R}^n[/itex] -> [itex]\mathbb{R}[/itex]. For every x [itex]\in[/itex] [itex]\mathbb{R}^n[/itex] we define the following:

    k(x) = max{f(x), g(x)}
    h(x) = min{f(x), g(x)}

    The question is:
    if lim x-> a k(x) exists and lim x-> a h(x) exists, and the limits are equal, does that imply that lim x->a f(x) exists?


    2. Relevant equations



    3. The attempt at a solution
    Suppose we define f(x) = 1/x and g(x) = x. At x = 0 f(x) is not defined, so g(x) is the minimum and the maximum at x = 0. Therefore k(x) = g(x) and h(x) = g(x). We know that lim x-> 0 h(x) exists and lim x-> 0 k(x) exists, but lim x -> 0 f(x) does not exists.

    Is my work here correct or am i wrong in assuming that if f(0) is not defined then g(0) is the maximum and the minimum at x = 0?
     
  2. jcsd
  3. Feb 26, 2012 #2

    SammyS

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    Hello Tomath. Welcome to PF !

    lim x→0 k(x) does not exist. Furthermore, this limit has nothing to do with whether or not k(0) exists, nor does it depend upon the value of k(x). From the left of x=0, k(x) approaches 0. From the right it approaches +∞ .
     
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