So, the problem that we were solving was
$$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$
To figure out whether the series converged or diverged, after simplification, I asked my professor whether finding the limit of the inside of the function to determine whether the inside function was divergent or convergent would be helpful. My logic was this: If it's divergent, infinity to the power of $n$ as $n$ approaches infinity is just infinity, and if it's between $-1$ and $1$ then it approaches a finite value, right? Meaning that, if we were to take the limit of the inside it would ultimately determine what the function did after taken to the power of n.
I was told that we simply couldn't do this but her explanation was a bit lackluster, it was basically "because I said so." Please tell me exactly how I'm wrong so I can better understand what I'm doing. Thank you!
Best Answer
In general we can't pull a limit past a variable that occurs in the limiting operation, because that might change the result. Here's an example which is easier than the problem you're working on:
$$ \lim_{x \to 1}\ (x-1)^{x-1} $$
If we take the limit of the inside function first, we'll get $0$, but in fact, the correct answer is $1$.
Now it is true that you can slide a limit operation past a variable that doesn't occur in the limit:
$$ \lim_{x \to 0} (x^y) = (\lim_{x \to 0} x)^y = 0^y = 0. $$
And it's true that you can slide a limit operation past a continuous function:
$$ \lim_{x \to \infty} \log(1 + \frac{1}{x}) = \log\left(\lim_{x \to \infty} 1 + \frac{1}{x}\right) = \log(1) = 0. $$
So it's easy to get tempted to think you can slide limits past whatever you want, but in fact you'll get the wrong answer.