TITLE: Finding the Limit of a Sequence (No L'Hopital's Rule)
QUESTION [18 upvotes]: Okay, I feel almost silly for asking this, but I've been on it for a good hour and a half. I need to find:
$$\lim_{n \to\infty}\left(1-\frac{1}{n^{2}}\right)^{n}$$
But I just can't seem to figure it out. I know its pretty easy using L'Hopital's rule, and I can "see", that it's going to be $1$, but apparently it's possible to compute using only the results I've proved so far. That is, the sandwich theorem, the addition/subtraction/multiplication/division/constant multiple limit laws and the following standard limits:
$\displaystyle\lim_{n \to\infty}\left(n^{\frac{1}{n}}\right)=1$
$\displaystyle\lim_{n \to\infty}\left(c^{\frac{1}{n}}\right)=1$, where $c$ is a real number, and
$\displaystyle\lim_{n \to\infty}\left(1+\frac{a}{n}\right)^{n}=e^{a}$.
As well as those formed from the ratio of terms in the following hierarchy:
$$1<\log_e(n)<n^{p}<a^{n}<b^{n}<n!<n^{n}$$
Where $p>0$ and $1<a<b$
e.g. $\displaystyle\lim_{n \to\infty}\left(\frac{\log_e(n)}{n!}\right)=0$
At first I thought maybe I could express the limit in the form of the $e^{a}$ standard limit, but I couldn't seem to get rid of the square on $n$. I've also tried the sandwich theorem, it's obviously bounded above by $1$, but I just couldn't find any suitable lower bounds. I'd really appreciate some help, or even just a little hint, thanks!

REPLY [1 votes]: Here's an argument that doesn't require any special concepts like Taylor Series or the limit definition of $e$:
$$\begin{align}
\lim_{n\to\infty} \left(1-\frac1{n^2}\right)^n
&= \lim_{n\to\infty} \left(1-\frac1n\right)^n\left(1+\frac1n\right)^n\\
&= \lim_{n\to\infty} \left(\frac{n-1}n\right)^n\times \lim_{m\to\infty}\left(\frac{m+1}m\right)^m\\
&= \lim_{n\to\infty} \left(\frac{n-1}n\right)^n\times \lim_{p\to\infty}\left(\frac{p}{p-1}\right)^{p-1}\\
&= \lim_{n\to\infty} \left(\frac{n-1}n\right)^n\left(\frac{n}{n-1}\right)^{n-1}\\
&= \lim_{n\to\infty} \left(\frac{n-1}n\right)\\
&= 1
\end{align}$$
Here, I've split the limit of the product into the product of limits, temporarily, using $m$ for the new variable. Then, I've let $m=p-1$, and then recombined the product.
The only thing this approach is conditional on is the convergence of the two separate limits, and may not even require that much.
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