Absolute Value Continuity

[Jason76]

Determine the continuity of \(\displaystyle M(x) = \dfrac{|x|}{x}(x^{2} - 1)\)

It's said that for \(\displaystyle x > 0\), \(\displaystyle |x| = x\) and for \(\displaystyle x < 0, |x| = -x\) but isn't the absolute value of a negative number positive?

Other questions:

\(\displaystyle M(x) = \dfrac{x}{x}(x^{2} - 1) = x^{2} - 1\) for \(\displaystyle x > 0\) Ok, makes sense.

\(\displaystyle M(x) = \dfrac{-x}{x}(x^{2} - 1) = -(x^{2} - 1) = 1 - x^{2}\) for \(\displaystyle x < 0\) - How did the arithmetic come out to that?

\(\displaystyle M(x) = \dfrac{x}{x}(x^{2} - 1) = x^{2} - 1 = \dfrac{0}{0} = undefined\) for \(\displaystyle x = 0\) But isn't zero on top and bottom indeterminate, while a 0 in the bottom is undefined?

 

[daon2]

It's said that for \(\displaystyle x > 0\), \(\displaystyle |x| = x\) and for \(\displaystyle x < 0, |x| = -x\) but isn't the absolute value of a negative number positive?

Yes. For example \(\displaystyle -7 < 0\) So \(\displaystyle |-7| = -(-7)\)

 

[Jason76]

Yes. For example \(\displaystyle -7 < 0\) So \(\displaystyle |-7| = -(-7)\)

Then the book must have a typo, but I've seen this stated in two books. The absolute value of -some-number should be some-number, not -some-number.

 

[daon2]

Then the book must have a typo, but I've seen this stated in two books. The absolute value of -some-number should be some-number, not -some-number.

It isn't a typo.

\(\displaystyle x=-7 \implies x<0 \implies |-7|=|x| = -x=-(-7)\)

\(\displaystyle M(x) = \dfrac{x}{x}(x^{2} - 1) = x^{2} - 1\) for \(\displaystyle x > 0\) Ok, makes sense.

\(\displaystyle M(x) = \dfrac{-x}{x}(x^{2} - 1) = -(x^{2} - 1) = 1 - x^{2}\) for \(\displaystyle x < 0\) - How did the arithmetic come out to that?

\(\displaystyle M(x) = \dfrac{x}{x}(x^{2} - 1) = x^{2} - 1 = \dfrac{0}{0} = undefined\) for \(\displaystyle x = 0\) But isn't zero on top and bottom indeterminate, while a 0 in the bottom is undefined?

I cannot assess this because I don't know what M(x) is. The third part might be saying the function M(x) is not defined at x=0. If it was defined one could write down the value.

edit: OK, I see it now in your post. They are just replacing x with -x, since if x<0, |x|=-x. If x <0, |x|/x = (-x)/x = -1, and then the negative is distributed to x^2-1