soft question - Why is 1 raised to infinity Not defined and not "1"

$1$ 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?

Answer

What $1^\infty$ is, or is not, is merely a matter of definition. Normally, one would only define $a^b$ for some specific class of pairs of $a,b$ - say $b$ - positive integer, $a$ - real number.

When extending the definition of exponentiation to more general pairs, the key thing people keep in mind is that various nice properties are preserved. For instance, for $b$ - positive integer, you want to put $a^{-b} = \frac{1}{a^b}$ so that the rule $a^ba^c = a^{b+c}$ is preserved.

It may make sense in some context to speak of infinities in the context of limits, but this is usually more a rule of thumb than rigorous mathematics. This may be seen as extending the rule that $(a,b) \mapsto a^b$ is continuous (i.e. if $\lim_n a_n = a$ and $\lim_n b_n = b$, then $\lim_n a_n^{b_n} = a^b$) to allow for $b_n \to \infty$. For instance, you may risk saying that:

$$\lim_{n} (2+\frac{1}{n})^n = 2^{\infty} = \infty$$
If you agree to use rules of this kind, you might be tempted to also say:
$$\lim_{n} (1+\frac{1}{n})^n = 1^{\infty} = 1$$
but this would lead you astray, since in reality:
$$\lim_{n} (1+\frac{1}{n})^n = e \neq 1$$
Thus, it is safer to leave $1^\infty$ undefined.

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A more thorough discussion can be found on Wikipedia.