1^infinity doesn't exist. The limit of 1^x as x approaches infinity is 1. 1 is being multiplied by itself infinitely, and the result is always one. I dunno why everyone keeps proving different things.
The problem is with limits like e=(1+1/x)^x. Here, you have a number that is approaching one being taken to the power of a number that is approaching infinity. This is the most common application of the scenario 1^infinity, in which the base does not actually equal 1 and the exponent does not actually equal infinity, but both numbers approach their respective goals. I believe this is the reason your teacher told you 1^infinity does not equal 1.
The reason this does not equal one is because any number that is greater or less than one will rapidly diverge from one. 1.000000000001^(10^20) is a very large number, even though the initial number was very small. 0.9999999999^(10^20) is a very small number that is close to zero, even though the initial number was very close to one. This is why the limit of any number that approaches one to the power of any number that approaches infinity is undefined, or in mathematically incorrect but simpler terms, 1^infinity is undefined.
The problem is with limits like e=(1+1/x)^x. Here, you have a number that is approaching one being taken to the power of a number that is approaching infinity. This is the most common application of the scenario 1^infinity, in which the base does not actually equal 1 and the exponent does not actually equal infinity, but both numbers approach their respective goals. I believe this is the reason your teacher told you 1^infinity does not equal 1.
The reason this does not equal one is because any number that is greater or less than one will rapidly diverge from one. 1.000000000001^(10^20) is a very large number, even though the initial number was very small. 0.9999999999^(10^20) is a very small number that is close to zero, even though the initial number was very close to one. This is why the limit of any number that approaches one to the power of any number that approaches infinity is undefined, or in mathematically incorrect but simpler terms, 1^infinity is undefined.