Giving infinity a value

1. Aug 24, 2015

Einstein's Cat

Is infinity divided by infinity equal to 1? 6 divided by 6 is equal to 1 however as infinity resembles 0 in the sense that 0 dived by 0 is equal to 0, I am uncertain whether infinity divided by infinity would equal 1 or instead, infinity.

2. Aug 24, 2015

DEvens

Usually you can't do things like multiply or divide by infinity. It is not defined. Similarly, 0 divided by 0 is not 0. It is not defined.

What you can sometimes do is examine a limit. So $\lim_{x->0} \frac{\sin(x)}{x}$ is defined, and is 1. So in this sense, in this case, dividing a zero by a zero gives you 1. But only as the limit.

https://en.wikipedia.org/wiki/L'Hôpital's_rule

3. Aug 24, 2015

soarce

The uncertainties of the type $\infty \cdot 0$, $\infty/\infty$ or $0/0$ acquire a definite value only as a limit, you can't simply operate with $\infty$ as being a number. So, unless we can go through the limit process it makes no sense to say that some uncertainty is equal to some value.

4. Aug 24, 2015

AliGh

You are considering infinity as a constant number
Depends on the infinities you are working with quotient of two infinities can be zero or infinity too

5. Aug 24, 2015

Einstein's Cat

6. Aug 31, 2015

Staff: Mentor

You have managed to pack a number of things that aren't true into a small number of words.

No.
The indeterminate form $[\frac{\infty}{\infty}]$ shows up in calculus as limits that can literally come out to any number, as well as negative or positive infinity. Here are some simple examples:
1. $\lim_{x \to \infty}\frac{x^2}{x} = \infty$
2. $\lim_{x \to \infty}\frac{x}{x^3} = 0$
3. $\lim_{x \to \infty}\frac{x^2 + 3}{3x^2 - x + 7} = \frac 1 3$

No, not at all.

No.
Division by 0 is not defined. The indeterminate form $[\frac 0 0]$ also shows up in calculus limits, and can come out to any number. Some examples of this:
1. $\lim_{x \to 0}\frac{x^2}{x} = 0$
2. $\lim_{x \to 0}\frac{x}{x^2}$ does not exist
3. $\lim_{x \to 0}\frac{x}{x^3} = \infty$
4. $\lim_{x \to 0}\frac{\sin(2x)}{x} = 2$