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Math Help - Evaluate limit of a absolute value

  1. #1
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    Evaluate limit of a absolute value

    lim as x goes to 0

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

    Something with piecewise?
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  2. #2
    MHF Contributor chisigma's Avatar
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    For 0<x<\frac{1}{2} is |2x-1|= 1-2x and |2x+1|= 2x+1 so that is...

    \displaystyle \lim_{x \rightarrow 0+} \frac {|2x-1|- |2x+1|}{x} =  \lim_{x \rightarrow 0+} \frac {-4x}{x} = -4 (1)

    For -\frac{1}{2}<x<0 is [also] |2x-1|= 1-2x and |2x+1|= 2x+1 so that is...

    \displaystyle \lim_{x \rightarrow 0-} \frac {|2x-1|- |2x+1|}{x} =  \lim_{x \rightarrow 0-} \frac {-4x}{x} = -4 (2)

    Kind regards

    \chi \sigma
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