# Thread: Analyzing a Limits Solution

1. ## 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)

2. ## 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?

3. ## 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