Is Using Absolute Value for Infinity Common Practice in Limits?

[blunkblot]

Hi,

I came across a book which looks at a problem like

$$\lim_{x \to 0}\frac{1}{x}$$

So you approach from 0-, and get -∞, approach from 0+, get ∞

Then it would write the answer as

$$\lim_{x \to 0}\frac{1}{x} = \left| \infty \right|$$

It looks bizarre to me. How do you parse this? Is this common practice or just bad notation?

 

[Landau]

Bad, uncommon, bizarre notation. I have never seen it before.

 

[Hurkyl]

If you use the projective reals instead of the extended reals, the limit exists. It's possible that that book uses that notation to indicate that it's using the infinite element of the former, rather than one of the two infinite elements of the latter. But I too have never seen that before.

 

[darkside00]

you can get negative infinite. Why you taking the absolute value of it?

Although, the slope will eventually equal the same as you approach 0 from either direction

 

[blunkblot]

Presumably the book is using the absolute value to indicate that the solution includes both -inf and +inf. That's how I read it, but I'm happy to know I'm not the only one who finds it bizarre.

 

[Landau]

What book is it?

 

[Gigasoft]

$$\lim_{x \to 0}\frac{1}{x} = \left|\infty\right|$$
This expression is blatantly incorrect and shows that the author has a basic misunderstanding of mathematics.

What he probably means, if you could write it like that, is:
$$\left| \lim_{x \to 0}{\frac{1}{x}}\right|=\infty$$

But you can't. An expression has one definite value, and $$\lim_{x \to 0}{\frac{1}{x}}$$ doesn't exist. One has to write:
$$\lim_{x \to 0^+}{\frac{1}{x}}=\infty$$
$$\lim_{x \to 0^-}{\frac{1}{x}}=-\infty$$

 

[Landau]

Or
$$\lim_{x\to 0}\left|\frac{1}{x}\right|=\infty$$
:)

 

[darkside00]

that works