Why can I not put the limit inside the limit definition of $e$? We know $e:= \lim_{n\to \infty}(1+\frac{1}{n})^n$
We also know $x^n$ is continuous. Why is there a contradiction in the following?
$e=\lim_{n\to \infty}(1+\frac{1}{n})^n=(\lim_{n\to \infty}(1+\frac{1}{n}))^n=1$
Is it because $x^n$ isn't really a continuous function, because the $n$ is not fixed but rather something like infinity?
 A: You can't move a limit that depends on $n$ "inside" the function if the outside function depends on $n$. For example, it is clearly true that
$$\lim_{n \to \infty} 2^n= \infty .$$
However, if you move the limit "inside," you get
$$\lim_{n \to \infty} 2^n= (\lim_{n \to \infty} 2)^n=2^n$$
which does not make much sense, since you do not know what $n$ is... $n$ was supposed to go to infinity. However, since you "got rid of" the limit, $n$ has no meaning anymore.
In other words, whenever you are calculating limits, $n$ should disappear at the same step as the words $\lim_{n \to \infty}$. If these two things don't go away at the same time, most likely it is a sign that you did something wrong.
I hope that helps!
A: The actual problem is you can't write $(\lim_{n\to\infty}f(n))^n$, or anything else that uses $n$ outside the limit, because it's a dummy variable that exists only inside the limit. The role of $\infty$ is irrelevant; you also can't, for the same reason, jump from $\lim_{x\to 2}(x+x)=4$ to writing $(\lim_{x\to 2}x)+x$.
A: Because the exponent is variable depending. In this case you get $1^{\infty}$ which is indeterminate, or means merely that this limit can be anything. 
