TITLE: Does $1^{\infty}=e$ or $1^{\infty}=1$?
QUESTION [5 upvotes]: In fact the real question is: Does $\lim\limits_{n\to\infty}1^{n}=e$?.
I know that 
$$
\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n=e,
$$
So, can we say that $1^\infty=e$?
And, by logic, This product 
$$\underset{\text{infinity times}}{1\cdot1\cdot\ldots\cdot1},$$
gives $1$ not other value.

REPLY [2 votes]: Congratulations, you have discovered a false belief.

False belief. Insertion of limits "rule" $$\lim_{x \rightarrow x_0}f(g(x),x) = \lim_{x \rightarrow x_0}f\left(\lim_{y \rightarrow x_0}g(y),x\right)$$

Counterexample.
$$f(a,b) = \left(1+a\right)^b \qquad g(a)=\frac{1}{a}$$
Then $$\lim_{x \rightarrow \infty}f(g(x),x) = e, \qquad \lim_{x \rightarrow \infty}f\left(\lim_{y \rightarrow \infty}g(y),x\right) = 1$$