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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Trigonometric functions}


\subsubsection{Limits of trig functions}

In order to understand the calculus of \( \sin (x) \), or any trigonometric
function, we first need to find some limits involving trigonometric functions.
First, we deal with the limit  \[
\lim _{x\rightarrow 0}\frac{\sin (x)}{x}.\]


To apply the squeeze principle to this situation, we need to look for functions
on either side, to compare it to.\footnote{%
I know, that is a dangling preposition. Too bad. As Winston Churchill once said
about such grammatical finery: {}``That is the sort of thing, up with which
I will not put{}''.
} The picture below shows what we need. 

\vspace{0.3cm}
{\par\centering \includegraphics{squeeze-triangles-3.ps} \par}
\vspace{0.3cm}

The area of the other triangle, \( OBD \), is \[
\frac{1}{2}\tan (\theta ),\]
 since it is a right triangle with base 1 and height \( \tan (\theta ) \),
which you can see by turning the picture upside down. 
\end{document}
