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Math Help - Finding a limit with absolute values

  1. #1
    Member Chokfull's Avatar
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    Finding a limit with absolute values

    My problem is

    \lim_{x\to0} \frac {|2x-1|-|2x+1|} {x}

    So if i set t=|2x-1| and c=|2x+1| i can get

    2x-1=\pm t and 2x+1=\pm c
    x=\frac {1\pm t} {2} and x=\frac {-1\pm c} {2}
    \frac {1\pm t} {2}=\frac {-1\pm c} {2}
    |2\pm t|=c

    then
    <br />
\frac {t-|2\pm t|} {(1\pm t)/2}

    and as x approaches 0, t approaches 1 so

    \lim_{t\to1} \frac {2t-2|2\pm t|} {1\pm t}

    since t approaches 1 and the denominator cannot be 0

    \frac {2-2|2\pm t|} {1\pm t=2}
    1-|1\pm 2|=0 or 2

    But the book says the answer is -4!!!!!!!
    anyone able to help?
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  2. #2
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     - .5 < x < .5\;\& \,\frac{{\left| {2x - 1} \right| - \left| {2x + 1} \right|}}{x} = \frac{{\left( { - 2x + 1} \right) - \left( {2x + 1} \right)}}<br />
{x}
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