Regarding limits and $1^\infty$ 
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Why is $1^{\infty}$ considered to be an indeterminate form 

I have some questions about limits and the undefinability of $1^\infty$.
For example, is $\lim_{x\to\infty}1^x$ indefinite? Why is it not $1$? Or do mathematicians, when saying that $1^\infty$ is indefinite, actually refer to cases such as $lim_{x\to\infty} \left(1 + \frac{a}{x}\right)^x$ where even though at a first glace the result is $1$, this is actually a special case and it is equal to $e^a$?

By itself, $1^{\infty}$ doesn't mean anything, except the implication that you had a quantity of the form $a^b$, and $\lim_{x\rightarrow\infty}a=1$ and $\lim_{x\rightarrow\infty}b=\infty$.  Depending on how quickly and from what direction $a$ approaches 1, and how quickly $b$ approaches $\infty$, $a^b$ could approach any non-negative number, infinity, or nothing in particular at all.
