Is Infinity Divided by Infinity Equal to 1?

[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.

 

[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

 

[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.

 

[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

 

[Einstein's Cat]

Cheers for all your help

 

[Mark44]

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.

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

Is infinity divided by infinity equal to 1?

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##

as infinity resembles 0

No, not at all.

0 dived divided by 0 is equal to 0

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##