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Show by means of example that lim/x→a/[f(x)g(x)] may exist

  1. Sep 18, 2013 #1
    Show by means of example that lim/x→a/[f(x)g(x)] may exist even though neither lim/x→a/f(x) nor lim/x→a/g(x) exists.

    I have tried using examples such as piecewise functions and rational functions, but can never validate the statement.

    Any guidance and help would be great.

    Thanks.
     
  2. jcsd
  3. Sep 18, 2013 #2

    mathman

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    Let f(x) be any crazy function. Let g(x) = 1/f(x).
     
  4. Sep 18, 2013 #3

    arildno

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    You forgot to insert Arildno's corollary:
    "Let f(x) be any crazy function. Let g(x) = 1/f(x). THEN, g(x) is most likely also a crazy function"

    Not very useful in this context, of course, but the result is beautiful, nonetheless. :smile:
     
  5. Sep 18, 2013 #4

    pasmith

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    Consider the function which is equal to 1 if its argument is rational and 0 otherwise.
     
  6. Sep 18, 2013 #5

    HallsofIvy

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    I presume you mean "let f(x)= 1 if x is rational, 0 if x is irrational.

    And then let g(x)= 0 if x is rational, 1 if x is irrational.


    fg(x)= 0 for all x so it trivially differentiable.
     
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