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This problem will walk through a proof of the trigonometric limit
\[ \lim_{x \rightarrow 0} \frac{\sin{x}}{x}  = 1\] using the squeeze theorem.
\begin{enumerate}
    \item Let $\theta$ be any angle between $0$ and $\frac{pi}{2}$. The picture below shows the unit circle with a few extra points and triangles drawn in. Find the area of each of the following regions:
    \begin{itemize}
        \item Triangle $OAB$
        \item Sector $OAB$
        \item Triangle $OCB$ (Hint: You'll need to use similar triangles for this part)
    \end{itemize}
    
    \begin{center}
        \begin{tikzpicture}
            \draw[black, thick, <->] (-.5, 0) -- (6.5, 0);
        \draw[black, thick, <->] (0, -.5) -- (0, 6.5);
        \draw[fill] (0,0) circle[radius=0.1] node[below left] {$O$};
        \draw (0,5) arc (90:0:5);
        \draw (0,0) -- (5, 3.75) -- (5,0);
        \draw (4,3) -- (5,0);
        \draw[dashed] (4,3) -- (4,0);
        \draw[fill] (4,3) circle[radius=0.1] node[left] {$A$};
        \draw[fill] (5,3.75) circle[radius=0.1] node[above] {$C$};
        \draw[fill] (5,0) circle[radius=0.1] node[below left] { $B$};
        \node at (1, 0.4) {\Large $\theta$};
        \end{tikzpicture}
    \end{center}
    \item Put these three areas in an order based on the way that the geometric objects contain each other.
    \item We want to use the squeeze theorem to say something about \[ \frac{\sin{\theta}}{\theta} \] as $\theta \rightarrow 0$. Rearrange the inequalities in (b) to give information about this ratio. 
    \item Take the limit as $\theta \rightarrow 0$ and use the squeeze theorem to get the desired result. 
\end{enumerate}

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