Why is $ 1^{\infty} \not= \lim_{t \to \infty} 1^t = 1$?

Question

Why is $ 1^{\infty} \not= \lim_{t \to \infty} 1^t = 1$?

It seems like they're saying the same thing. After all, $1$ to the power of any positive integer is equal to $1$.

I suspect that I'm misunderstanding some fundamental concept and would appreciate it if someone could explain it to me.

Thank you.

Answer 1

The '$1$' in $1^\infty$ doesn't represent the number one; rather, it's a placeholder for any sequence whose limit is $1$. In this sense, it's similar to the zeroes in the indeterminate form $\dfrac00$; those aren't literal zeroes, but they're stand-ins for an expression whose limit is zero. The classic example of a $1^\infty$ form, of course, is the exponential limit itself: $\lim_{n\to\infty}(1+\frac1n)^n$. It's true that $\lim_{n\to\infty}(1+\frac1n)=1$, of course, and it's true that $\lim_{n\to\infty}1^n=1$; but we can't commute a 'partial' limit and say that if $L=\lim_{n\to\infty}a_n$ then $\lim_{n\to\infty}a_n^{b_n}=\lim_{n\to\infty}L^{b_n}$.

Answer 2

There is an indeterminate form called $1^\infty$. We say it is indeterminate despite $$ \lim_{t\to\infty} 1^t = 1, $$ which is correct, but there are other cases like $$ \lim_{x \to \infty}\left(1+\frac{1}{x}\right)^x $$ which have the form $1^\infty$ but do not converge to $1$.