# Absolute value of infinity

1. Jun 10, 2010

### 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?

2. Jun 10, 2010

### Landau

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

3. Jun 10, 2010

### Hurkyl

Staff Emeritus
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.

4. Jun 10, 2010

### 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

Last edited: Jun 10, 2010
5. Jun 10, 2010

### 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.

6. Jun 11, 2010

### Landau

What book is it?

7. Jun 11, 2010

### 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:
$$\lim_{x \to 0^+}{\frac{1}{x}}=\infty$$
$$\lim_{x \to 0^-}{\frac{1}{x}}=-\infty$$

8. Jun 12, 2010

### Landau

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

9. Jun 14, 2010

that works