\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}