I can't find the limit absolute value function.

In summary, the conversation is about finding the limit of the absolute value function (|x-2|-2)/x as x approaches 1. The final answer is -1, but there was confusion about how to arrive at this answer. After discussing different approaches and clarifying the concept of limits, it was determined that the simplified formula for f(x) when x is near 1 is f(x)=-1.
  • #1
Vola
Gold Member
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How find the limit of absolute value function?

Hi everybody,

I can't find the limit of (abs(x-2)-2)/x as x-->1.
I know it's (-1) but I don't see how you get to it.
If I take (x-2)>0 I get L=-3,(x-2)<0 I get L=-1.
However according to the graph two sided limit does exist and it's (-1).

Please help
Thanks
 
Last edited:
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  • #2
Vola said:
Hi everybody,

I can't find the limit of (abs(x-2)-2)/x as x-->1.
I know it's (-1) but I don't see how you get to it.
If I take (x-2)>0 I get L=-3,(x-2)<0 I get L=-1.
However according to the graph two sided limit does exist and it's (-1).

Please help

What happens when x is near 1?

RGV
 
  • #3
is it just a "plug-in" problem?

(|1-2|-2)/1 => (|-1|-2)/1 => (1-2)/1 =-1?
 
Last edited:
  • #4
Vola said:
is it just a "plug-in" problem?

(|1-2|-2)/1 => (|-1|-2)/1 => (1-2)/1 =-1?

What is your answer to the question I asked you?

RGV
 
  • #5
The value of function is negative(?)
 
  • #6
Vola said:
The value of function is negative(?)

Let me ask it one last time: if f(x) = (|x-2|-2)/x, what would be the simplified formula for f(x) when x is near 1? Don't guess---calculate!

RGV
 
  • #7
f(x)=-1
 
  • #8
Good! For all x less than 2, f(x)= -1 so the limit at any x less than two is -1.

If I take (x-2)>0 I get L=-3,(x-2)<0 I get L=-1.
But you don't "take (x- 2)> 0"! That would be x> 2 and you want the limit at x= 1. You can't "go to 1" if keep x> 2.
 
  • #9
thank you HallsofIvy and Ray
 

FAQ: I can't find the limit absolute value function.

What is a limit in a mathematical context?

A limit is a value that a function or sequence approaches as the input or index approaches a certain value. It is often used to describe the behavior of a function near a certain point.

How do you find the limit of an absolute value function?

To find the limit of an absolute value function, you need to determine the left and right limits separately. For the left limit, plug in values that approach the critical point from the left, and for the right limit, plug in values that approach the critical point from the right. If the left and right limits are equal, then that value is the limit of the function.

Can the limit of an absolute value function be undefined?

Yes, the limit of an absolute value function can be undefined if the left and right limits are not equal. This usually happens when the function has a sharp turn or jump at the critical point.

Are there any special cases when finding the limit of an absolute value function?

Yes, there are two special cases to consider when finding the limit of an absolute value function. The first is when the critical point is at 0, in which case the left and right limits will always be equal. The second is when the critical point is at a point where the function is not defined, in which case the limit will also be undefined.

How can knowing the limit of an absolute value function be useful?

Knowing the limit of an absolute value function can help determine the behavior of the function near a certain point. It can also be useful in finding the derivative of the function, as the limit is used in the definition of the derivative.

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