calculus - $lim_{ntoinfty} 1^n = 1$?

Is it really true that

$$\lim_{n\to\infty} 1^n = 1$$

?

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:

$$\lim_{n\to\infty}\left(\color{blue}{1+\frac{1}{n}}\right)^\color{red}{n} = e \ne1$$ where 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.