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Math Help - Analyzing a Limits Solution

  1. #1
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    Analyzing a Limits Solution

    Greetings all,

    I am reviewing a limit question for a test and am trying to figure out the reason why it was solved as it was. Here is the question:

    lim
    x->4+

    (4-x)|3x-14|
    ___________
    |4-x|

    Solution:
    since x>4
    (4-x)<0 (as x>4)
    |3x-14|= 14-3x
    |4-x|=x-4

    (4-x) |3x-14|
    ___________
    (x-4)

    = -(14-3x)
    =-2

    I am confused as to why the |3x-14| becomes 14-3x. If the x is to be greater then 4, would |3x-14| be positive?
    So in future questions with absolute values, should I first rewrite the absolute value (like in the x-4 case) so that it matches the situation (i.e. if x is less then or greater then a number, write it so that it factors that in?...is there an easier way to say that? :P)
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  2. #2
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    Re: Analyzing a Limits Solution

    I think for |3x-14|, if x is subbed in, the value is positive but if 5 is subbed into 14-3x, then the value becomes negative.
    I think it makes sense if subbing 4, in both cases the value is positive. I think this is why I am getting confused. I am to use values greater then 4. How do you treat these cases? Sub with the actual limit, even though it wouldn't work with other numbers?
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  3. #3
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    Re: Analyzing a Limits Solution

    If you substitute 4, 3x - 14 = 3(4) - 14 = 12 - 14 = -2, not positive. That's why absolute value becomes the negative of it. It is not positive, and you do not need to go till 5 to check the sign, any neighborhood of 4 would be good, which means, usually just 4 is good.

    Salahuddin
    Maths online
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