Is it really true that
lim
?
We were taught throughout our entire math courses that [1^{\infty}] is an undetermined form....Am I missing something here?
Answer
Is it really true that
\lim_{\color{red}{x}\to\infty} 1^n = 1
You probably mean:
\lim_{\color{blue}{n}\to\infty} 1^n = 1
and yes, this is true because 1^n = 1 for all n.
The expression "1^{+\infty}" is indeterminate and the limit above doesn't contradict that.
Perhaps you know the following well-known limit too:
\lim_{n\to\infty}\left(\color{blue}{1+\frac{1}{n}}\right)^\color{red}{n} = e \ne1where you also have \color{blue}{1+\frac{1}{n}}\to 1 and \color{red}{n}\to+\infty.
Combining both limits shows that you can have sequences c_n = \left(a_n\right)^{b_n} where a_n \to 1 and b_n\to+\infty but with different limits for c_n; which is why we call "1^{+\infty}" indeterminate.
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