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Question about absolute value limits?

  1. Jan 20, 2014 #1
    When do you check the limit from the right and left of a limit with an absolute value in the numerator or denominator?

    For example, why do you check the limit from both sides of:

    Lim x -> 3/2 (2x^2-3x)/absolute value(2x-3)

    But only the left side of:

    limit as x approaches -2

    (2-absolute value x)/(2+x)
     
  2. jcsd
  3. Jan 20, 2014 #2

    Office_Shredder

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    If the thing inside the absolute value is equal to zero when you plug in the limiting value of x, then depending on whether x is a little bigger or a little smaller it will change how the absolute value acts on the expression inside of it. For example

    [tex] \lim_{x\to 3/2} \frac{...}{|2x-3|} [/tex]
    if x = 3/2, 2x-3 = 0. So if x > 3/2, |2x-3| = 2x-3, but if x < 3/2, |2x-3| = 3-2x and you need to be careful about this distinction.

    On the other hand, if x = -2, 2-x = 4. So if x > -2 by a little bit, |2-x| = 2-x, but if x < -2 by a little bit, |2-x| = 2-x still. In both cases you get the same expression when you drop the absolute value sign.
     
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