Significance of absolute value bars in a limit property

[meer]

Solution to a limit problem can be written in an alternative form (property) as following:

limx->a f(x) = L if and only if limx->a |f(x) – L|= 0

(ref: alternative form property Mustafa Munem Calculus > section 1.8 > page 64)
I don’t understand the significance and necessity of absolute value bars here because in the following example I get the alternative form equation satisfied on both LHS and RHS.

 

[Steven G]

Did you try proving that theorem without the absolute value bars?? An example is not a proof! In theory, there can be examples where the absolute values bars are needed!

Having said that, I think that you are correct that the absolute value bars are not necessary.

Please use equal signs and not implication symbols!

 

[Zermelo]

Solution to a limit problem can be written in an alternative form (property) as following:

limx->a f(x) = L if and only if limx->a |f(x) – L|= 0

(ref: alternative form property Mustafa Munem Calculus > section 1.8 > page 64)
I don’t understand the significance and necessity of absolute value bars here because in the following example I get the alternative form equation satisfied on both LHS and RHS.

View attachment 34452

For real numbers x and y, the distance (metric) between them is |x - y|. Take a few examples and see if it holds up, take 2 points on the number line, measure the distance between them. It will be the absolute value of their difference.
In 2D space, we have a different metric, and you calculate the distance between points using the Pythagorean theorem. But in 1D, this |x - y| is enough.
So what the expression on the right is saying is “the distance between the number L and the values of the function f becomes small (tends to 0) when x->a”
So it doesn’t matter if L is to the right or to the left of f(x), the only thing that matters is that their distance is becoming small as x->a.

And I agree with Steven, the thing you did doesn’t make much sense, you already calculated the limit L on the left side, ofcourse if you subtract it from lim f(x) that you’ll get zero ?
a-a=0, right?

And I don’t believe you would use the “alternate form” to actually compute limits, but it’s important to understand what it says