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\begin{frame}
\frametitle{Finding a Tangent}

We move $Q$ closer and closer to $P$.
\begin{center}
\scalebox{.7}{
\begin{tikzpicture}[default]
\diagram{-.5}{6}{-.5}{4}{1}
\diagramannotate

\draw[name path=x2,ultra thick,cgreen] plot[smooth,domain=-0:6,samples=20] function{3-(x-2)**2 + 0.12*(x-1)**3} node[right] {$f(x) =x^2$};

\node[dot] (P) at (1,{2}) {};
\node[anchor=north west,at=(P.south)] {$P$};

\foreach \x in {4.5,3.5,2.5} {
\node[dot] (Q) at (\x,{3-pow(\x-2,2) + 0.12*pow(\x-1,3)}) {};
\node[anchor=south east,at=(Q.north)] {$Q$};
\through[opacity=.5,red,ultra thick]{2cm}{2cm}{P}{Q}
}

\tangent[cblue,ultra thick]{55}{50}{3-pow(\x-2,2) + 0.12*pow(\x-1,3)}{1}
\node[cblue] at (2.2,4.3) {tangent};
\end{tikzpicture}
}
\end{center}
\pause
The limit is the tangent.
\pause

\begin{block}{}
The \emph{tangent line} to the curve $f(x)$ at point $P = (a,f(a))$
is the line through $P$ with slope
\begin{talign}
m = \lim_{x\to a} \frac{f(x) - f(a)}{x - a}
\end{talign}
provided that the limit exists.
\end{block}
\end{frame}