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1 to the power infinity
If you calculate with infinity you should always be careful, as
1∞ = ?
So the result is not 1. How is that possible?
Explanation
You might not expect that. But the number 1 has special properties, and infinity is not a number. Therefore, you may not just conclude that
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It depends on how the number 1 and how infinitely came into being in the course of a calculation.
Example 1
We start from the composite function
where f (x) = 1 and g (x) = x. For x→∞ the following occurs
Now we take the same function, but with and g (x) = x this gives
This is the base e of the natural logarithm. By choosing other functions you can get different results. Therefore 1∞ only has a meaning if its origin is known. So in general applies
1∞ = ?