\begin{frame} \frametitle{The Limit of a Function} We investigate the function $x^2-x+1$ for values of $x$ near $2$. \bigskip \begin{minipage}{.49\textwidth} \scalebox{.7}{ \begin{tikzpicture}[default] \diagram{-1}{5}{-1}{6}{1} \diagramannotate \draw[ultra thick,cgreen] plot[smooth,domain=-1:2.85,samples=20] function{x**2-x+1} node[right] {$x^2-x+1$}; \node[dot] (P) at (2,{2^2-2+1}) {}; \end{tikzpicture} } \end{minipage} \begin{minipage}{.49\textwidth} \pause from below ($x < 2$):\\[.2ex] \scalebox{.9}{\small \begin{tabular}{|l|l|} \hline $x$ & $f(x)$ \\ \hline $1$ & $1$ \\ \hline $1.5$ & $1.75$ \\ \hline % $1.8$ & $2.44$ \\ % \hline $1.9$ & $2.71$ \\ \hline $1.99$ & $2.9701$ \\ \hline $1.999$ & $2.9970$ \\ \hline \end{tabular} } \medskip\pause from above ($x > 2$):\\[.2ex] \scalebox{.9}{\small \begin{tabular}{|l|l|} \hline $x$ & $f(x)$ \\ \hline % $3$ & $7$ \\ % \hline $2.5$ & $4.75$ \\ \hline $2.2$ & $3.64$ \\ \hline $2.1$ & $3.31$ \\ \hline $2.01$ & $3.0301$ \\ \hline $2.001$ & $3.0030$ \\ \hline \end{tabular} } \end{minipage} \medskip\pause From the tables we see: as $x$ approaches $2$, $f(x)$ approaches $3$. \begin{talign} \lim_{x\to 2} (x^2 - x + 1) = 3 \end{talign} \end{frame}