Why is 1^infinity indeterminate form and not just 1? Isn’t one to any power going to be 1?


Why is 1^infinity indeterminate form and not just 1? Isn’t one to any power going to be 1?

In: Mathematics

1 raised to the power of any *number* is 1. But in the real number system, infinity is not a number, and cannot be used in any arithmetic operations. It can be used as part of the notation for a limit, but you can’t perform calculations on it as if it were a number.

If it’s really 1 then yes, but this is when talking about limits, i.e. the 1 and infinity are are both limits of infinite sequences (or functions). In this case 1^infinity is shorthand for the limit of a(n)^b(n) where a(n) and b(n) are both infinite sequences, whose limits are 1 and infinity respectively.

For example, take a(n) = (1+1/n) and b(n) = n. In this example, the limits of a(n) and b(n) are 1 and infinity, but the limit of a(n)^b(n) = (1+1/n)^n is e (euler’s constant). On the other hand if b(n)=2n then the limit of a(n)^b(n) is e^(2).

1^infinity is just 1 as you say.

If you plot y = 1^x, you get a straight line which is the same as y = 1

Two possibilities for controversy.

* You can’t really plug ‘infinity’ into any equation. All you can say is that a function goes towards a limit as some value goes to infinity. lim x->infinity of 1^x is 1

* Are we really talking about exactly 1? Or is it something which goes towards 1? Say (1 + 1/x)… that goes towards 1 as x -> infinity but it’s not *exactly* 1. So if you have something like (1 + 1/x)^x and you look at how that behaves as x -> infinity. Then that won’t 1. That number will be *e* or ~2.718… But if it’s *exactly* one, then any power of it will also be 1.

The indeterminant form 1^infinity isn’t the same thing as 1^infinity. Indeterminant forms are the results of limits. It’s not that your base is 1 and your exponent is infinity. It’s that your base is headed towards 1 and your exponent is headed towards infinity. You get an indeterminant from when the result of your limit doesn’t actually tell you where the result is going, and this is one of those times.

As your base trends towards 1 that means that your overall result trends towards 1, because 1 to any exponent is 1. Your exponent increasing to infinity means your result trends towards infinity or 0 (depending on which direction the base is heading towards 1), because something to a high exponent gets really big or really small. So the base and the exponent are pulling your result in two different directions, since you don’t know which one wins the result is indeterminant.

In order to actually know the answer you would need to look at the original limit and find out if your base is going towards 1 faster than your exponent is going towards infinity.