Could $\infty$ hurt $1$ ?

Clearly, $1^1 = 1, 1^2 = 1$ , Easily can be proved by induction that if $1^n = 1$ then $1^{n + 1} = 1, \space \forall n \ge 1, n \in \mathbb{N}$ . This means the sequence $\{1^n\}_{n = 1}^{\infty}$ is the constant sequence $\displaystyle \{1\} \Rightarrow \lim_{n\to\infty} 1^n = 1 = N$

I think that is undefined... Please check your question again

The basic use of limits is when the expression is indeterminate. The given expression is actually not indeterminate as the 1 used in base is exact. Exact 1 to the power infinity is 1, as likely as exact 0 to the power 0 is 0 and 0 to the power exact 0 is 1. Normal 0 means tending to 0, and it is $not$ equal to 0.