TITLE: Why is 1 raised to infinity Not defined and not "1"
QUESTION [9 upvotes]: $1$ square is $1$, so is raised $1$ to $123434234$.
My maths teacher claims that $1$ raised to infinity is not $1$, but not defined. Is there any reason for this?
I know that any number raised to infinity is not defined, but shouldn't $1$ be an exception?

REPLY [3 votes]: By the way, just to give a much simpler answer (which indeed does not really explain the issue but might help if you're not studying calculus yet):
The problem here is that, in reality, $\infty$ is not a number. It is used to represent an unimaginably big number, but you obviously can't tell which. Therefore, infinity itself is not a defined number.
That's the background reason of why having $1^a$ with $a$ being a single, precise number does give you
$$
1^a=1
$$
while with $a=\infty$, or any other undefined number, it generally gives you $undefined$.
It's not the only case, actually. This is one example of indeterminate forms. Other similar examples are $\infty^0$ and $0*\infty$, which are also undefined.
To fully demonstrate it or show practical examples, you would actually have to go into functions and theory of limits (which are, by the way, not the actual values that functions take but where they converge, or approach, to). And the other answers have already done that, so I'll stop here.