limits - 1 to the power of infinity, why is it indeterminate?

I've been taught that $1^\infty$ is undetermined case. Why is it so? Isn't $1*1*1...=1$ whatever times you would multiply it? So if you take a limit, say $\lim_{n\to\infty} 1^n$, doesn't it converge to 1? So why would the limit not exist?

Answer

It isn’t: $\lim_{n\to\infty}1^n=1$, exactly as you suggest. However, if $f$ and $g$ are functions such that $\lim_{n\to\infty}f(n)=1$ and $\lim_{n\to\infty}g(n)=\infty$, it is not necessarily true that

$$\lim_{n\to\infty}f(n)^{g(n)}=1\;.\tag{1}$$

For example,

$$\lim_{n\to\infty}\left(1+\frac1n\right)^n=e\approx2.718281828459045\;.$$

More generally,

$$\lim_{n\to\infty}\left(1+\frac1n\right)^{an}=e^a\;,$$

and as $a$ ranges over all real numbers, $e^a$ ranges over all positive real numbers. Finally,

$$\lim_{n\to\infty}\left(1+\frac1n\right)^{n^2}=\infty\;,$$

and

$$\lim_{n\to\infty}\left(1+\frac1n\right)^{\sqrt n}=0\;,$$

so a limit of the form $(1)$ always has to be evaluated on its own merits; the limits of $f$ and $g$ don’t by themselves determine its value.