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Limit as x approaches 0 for given function

  1. Aug 5, 2014 #1
    1. The problem statement, all variables and given/known data

    lim x --> 0 for the function sqrt((1/x^2)-(1/x)) - sqrt((1/x^2)-(1/x))

    Analyze the cases x > 0 and x < 0

    3. The attempt at a solution

    The solutions book simplifies the expression to

    2x/(abs(x)*(sqrt(1+x)+sqrt(1-x)))

    I know how to evaluate the limit from here. But I'm wondering why did they only place the absolute value sign for the single x in the denominator that multiplies with this (sqrt(1+x)+sqrt(1-x)) expression. Why isn't the absolute value sign around every single x?


    The answer is that the limit doesnt exist.
     
    Last edited: Aug 5, 2014
  2. jcsd
  3. Aug 5, 2014 #2
    It looks like maybe your problem statement has a typo, but that doesn't seem to matter to the question that you asked.

    The absolute value comes into play because ##\sqrt{x^2}=|x|##.
     
  4. Aug 5, 2014 #3

    mfb

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    Why do you expect one?
     
  5. Aug 5, 2014 #4
    Ohh right I forgot out that equality. So you're saying all they did was sub in abs(x) wherever there as x^2?
     
  6. Aug 5, 2014 #5

    Ray Vickson

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    Your question must have a "typo" in it; what you have written is ##\sqrt{f(x)} - \sqrt{f(x)}##, which is identically 0 for all positive values of ##f(x) = 1/x^2 - 1/x.##
     
  7. Aug 5, 2014 #6

    mfb

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    It just appears once. They replaced ##\sqrt{x^2}## by abs(x) (note the square root), because ##\sqrt{x^2} = |x|##.
     
  8. Aug 5, 2014 #7
    No. I'm saying that they used the equality, ##\sqrt{x^2}=|x|##, as part of rewriting the given algebraic expression in way that is more easily analyzed in the context of this problem.

    One could say that they replaced certain instances of ##\sqrt{x^2}## with instances of ##|x|##.

    I don't like the word "substitute" to describe what is going on here. I can't exactly put my finger on why I don't like it. It just seems like the wrong word to use.
     
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