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Math Help - limit involving absolute values

  1. #1
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    Unhappy limit involving absolute values

    okay so im in calculus 2 right now, and im going over all my notes from cal 1 and i cant seem to figure this out.

    lim x->0- {abs(7x-2)-abs(7x+2)} / x

    i hate summer for this reason.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by darcyrenee08 View Post
    okay so im in calculus 2 right now, and im going over all my notes from cal 1 and i cant seem to figure this out.

    lim x->0- {abs(7x-2)-abs(7x+2)} / x

    i hate summer for this reason.
    Note that \left|7x-2\right|=\left\{\begin{matrix}7x-2 & \text{when }x\geq\frac{2}{7}\\-7x+2 & \text{when }x<\frac{2}{7}\end{matrix}\right. and \left|7x+2\right|=\left\{\begin{matrix}7x+2 & \text{when }x\geq-\frac{2}{7}\\-7x-2 & \text{when }x<-\frac{2}{7}\end{matrix}\right.

    So when we approach 0 from the left, we take \left|7x-2\right|=-7x+2 and \left|7x+2\right|=7x+2.

    Thus, we have

    \lim_{x\to0^-}\frac{\left|7x-2\right|-\left|7x+2\right|}{x}=\lim_{x\to0^-}\frac{-7x+2-(7x+2)}{x}=\lim_{x\to0^-}\frac{-7x+2-7x-2}{x} =\lim_{x\to0^-}\frac{-14x}{x}=-14
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    THANKS!!
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