calculus - How come $1^{infty}$ = undefined, while $2^{infty} = infty$ and $0^{infty} = 0$?

$1^\infty$ = undefined

$2^\infty = \infty$

$0^\infty = 0$

Why is $1^\infty$ undefined? People were trying to explain to me that infinity isnt part of the Real numbers, yet, $2^\infty$ and $0^\infty$ somehow ARE defined?

In my opinion $1^\infty = 1$. I mean isn't it not easy to prove since $1\times 1\times 1 \times \cdots=1$ no matter how many times you do it?