I can't find the limit absolute value function.

Vola

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

Ray Vickson

What happens when x is near 1?

RGV

Vola

is it just a "plug-in" problem?

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

Ray Vickson

What is your answer to the question I asked you?

RGV

Vola

The value of function is negative(?)

Ray Vickson

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

Vola

f(x)=-1

HallsofIvy

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

Vola

thank you HallsofIvy and Ray