\documentclass{beamer}
\usepackage{hyperref}  %Use this for referencing other pages and linking to them
\usepackage{colortbl}  %Use to unconver columns in a table

\usetheme{Warsaw}

\title[A treatise on $e$]{A Rational Limit:\\One more thing every Math Major should know}
\author{Barry Balof}
\institute{Math 497}
\date{Spring 2016}
\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}
\frametitle{A Useful Limit}


Somewhere in your mathematical past, you've encountered the following formula for $e$:

\begin{block}{`Definition' of $e$}
$$e=\lim_{n\to \infty} \left( 1+ \frac{1}{n} \right) ^ n$$
\end{block}
We seek to prove the formula using L'H\^{o}spital's Rule and some trickery.   

\end{frame}


\begin{frame}
\frametitle{Proving the Limit}

Rather than looking at 
$$\lim_{n\to \infty} \left( 1+ \frac{1}{n} \right) ^ n,$$

we will look at 

$$\lim_{n\to \infty} \ln\left(\left( 1+ \frac{1}{n} \right) ^ n\right)=\lim_{n\to \infty} n* \ln\left( 1+ \frac{1}{n} \right)$$


\end{frame}

\begin{frame}
\frametitle{An Indeterminate Form}

We note that as $n$ gets large, our limit is of the form $\infty*0$.  We make a rearrangement to use L'H\^{o}spital's Rule. 

$$\lim_{n\to \infty} n* \ln\left( 1+ \frac{1}{n}\right) $$
$$=\lim_{n\to\infty} \frac{\ln\left( 1+ \frac{1}{n}\right)}{\frac{1}{n}}\to \frac{0}{0}$$

\end{frame}

\begin{frame}
\frametitle{Using L'H\^{o}spital}

We now differentiate our numerator and denominator to obtain our limit.  

$$\frac{\ln\left( 1+ \frac{1}{n}\right)}{\frac{1}{n}}$$
$$\xrightarrow{\frac{f'}{g'}} \frac{ \frac{1}{1+\frac{1}{n}}*\frac{-1}{n^2}}{\frac{-1}{n^2}}$$
$$=\frac{1}{1+\frac{1}{n}}$$

\end{frame}

\begin{frame}
\frametitle{Taking the Limit}
Since 
$$\lim_{n\to\infty} \frac{1}{1+\frac{1}{n}} = 1,$$
We have that $$\lim_{n\to\infty} \frac{\ln\left( 1+ \frac{1}{n}\right)}{\frac{1}{n}}=1.$$
Hence,
$$\lim_{n\to \infty} \left( 1+ \frac{1}{n} \right) ^ n=e^1=e$$ 



\end{frame}


\end{document}