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Math Help - Sign Functions and Limits Problem

  1. #1
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    Sign Functions and Limits Problem



    The sign function sgn(x) is defined as follows:

    sgn(x)={x/lxl if x does not equal 0; 0 if x=0}

    Use the sign function to define two functions f and g whose limits as
    x -> 0 does not exist, but such that

    (a) lim [f(x)+g(x)] does exist
    x->0
    (b) lim (f(x))(g(x)) does exist
    x->0
    Please show work too, itll be very helpful!
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  2. #2
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    The easiest way of obtaining this would be for the one-sided limits to be different for both functions:

     f(x) =  sgn(x)
     g(x) = (-1) * sgn(x)

    Then:

     (f*g)(x) = (-1) * sgn^2(x) = -1
     (f+g)(x) = (-1) * sgn(x) + sgn(x) = (-1+1)*sgn(x) = 0

    But,

     \lim_{x\to0^+} f(x) = \lim_{x\to0^+} sgn(x) = 1

     \lim_{x\to0^-} f(x) =\lim_{x\to0^-} sgn(x) = -1

    Therefore  \lim_{x\to0^+} f(x) \neq \lim_{x\to0^-} f(x) \Rightarrow f(x) does not converge at x=0 and neither does g(x) in the same way.
    Last edited by Defunkt; August 24th 2009 at 01:20 PM.
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  3. #3
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    Sweet
    Thanks for your help
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  4. #4
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    Welcome!
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