<s>[INST] How to find $\lim\limits_{n\to \infty }(1 + \frac{1}{n})^n$ $\lim _{n\to \infty }\left(1 + \frac{1}{n}\right)^n$
How do I get started with this one? Variable substitution would be one way, but our lecturer hasn't covered that yet, so there should be some other way.
Usually with these kinds of limits we modify the function so that it resembles one of the standard limits that we may use without proving them.
 [/INST] it is known that the limit exists and is $e$. here is one way show that the limit exists and is between $2$ and $3.$ we will use the facts $(1 + \frac{1}{n})^n$ is an increasing sequence and it is bounded above by $3.$  with that the cliam will be established.
that $a_n=(1 + \frac{1}{n})^n$ is increasing follows from
$\dfrac{a_{n+1}}{a_n} = \left(\dfrac{n+2}{n+1}\right)\left( \dfrac{n^2 + 2n}{n^2 + n}\right)^n > 1$ for all $n \ge 1.$
 and the boudedness follows from
$\begin{align}(1 + \frac{1}{n})^n &= 1 + 1 + \dfrac{(1-1/n)}{2} + 
\dfrac{(1-1/n)(1-2/n)}{2*3} + \dfrac{(1-1/n)(1-2/n)(1-3/n)}{2*3*4}+ \cdots\\
& \le 1 + 1 + \dfrac{1}{2} + \dfrac{1}{2*3} + \dfrac{1}{2*3*4} + \cdots\\
& < 1 + 1 + \dfrac{1}{2} + \dfrac{1}{2*2} + \dfrac{1}{2*2*2} + \cdots = 3\\
\end{align}$
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