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Thread: 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 $\displaystyle 0<x<\frac{1}{2}$ is $\displaystyle |2x-1|= 1-2x$ and $\displaystyle |2x+1|= 2x+1$ so that is...

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

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

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

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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