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The reason why $1^\infty$ is indeterminate, is because what it really means intuitively is an approximation of the type $ (\sim 1)^ {\rm large , number}$. And while $1$ to a large power is 1, a number very close to 1 to a large power can be anything.....
Oh, it sounds like Ivm asking those questions rhetorically. I'm not. They are very subtle and difficult questions. Ponder them and think about them. Part of the problem is that we really abuse what $\sqrt {a} $ means. What does it mean? how many values does it have? Is it unambiguous in meaning?