What is the highest number you can state?

[Urwumpe]

If you can say that ∞/2 = ∞, then I think you can also say 1/∞ = 0 and ∞^-∞ = 0.

The problems start, when you try to make applied math with such phenomena. That's why it is possible to have infinity in limit functions (as these show a trend), but not possible to get clear results with such terms.

Just imagine the tangents function.

 

[agentgonzo]

Wrong... slightly. 1 / 0 or 1 / ∞ are not defined, but you can calculate the limit of a function x^-x

Not necessarily. If you are working in the field R \ {∞} then 1/0 is not defined (and 1 / ∞ cannot exist as you have explicitly forbidden it). If you are working over the set R ∪ {∞} then 1/0 is a valid operation and results in ∞.

 

[Jarvitä]

The largest number is about 45,000,000,000 although mathemeticians suspect there may be even larger numbers: 45,000,000,001?

Look around you. Just look around you!
:rofl:



Slightly more on-topic, the volume of the observable universe, measured in cubic Planck length units, to the hyper-4 iterated exponential of the height of itself.

 

[agentgonzo]

What about...
∞
∞
?

∞/∞ is undefined. Think of the sequence of 2x/x as x --> ∞. This is always equal to 2 and equal to 2 in the limit x --> ∞, therefore you could say that ∞/∞ = 2. However, apply the same logic to 3x/x (=3) and you get ∞/∞ = 3. That's why ∞/∞ is undefined.

Or sin(∞)

sin(∞) is also undefined as sin(x) is not a convergent function for unbounded x. Therefore taking the limit as x-->∞ of sin(x) is meaningless and therefore undefined.