Why is 1^infinity indeterminate form and not just 1? Isn’t one to any power going to be 1?
In: Mathematics
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.
Latest Answers