\begin{frame} \frametitle{Precise Definition of Limits} \precisebook \bigskip \emph{Alternative geometric interpretation:} \medskip \begin{minipage}{.49\textwidth} \scalebox{.9}{ \begin{tikzpicture}[default] {\def\diaborderx{1cm} \diagram{-.5}{3}{-.5}{3}{1}} \diagramannotatez \onslide<2->{ \draw[fill=cred,opacity=.2,draw=none] (-.25,1) rectangle ++(3.5,1); \draw[cred] (-.25,1) -- node[at start,left] {$L-\epsilon$} ++(3.5,0); \draw[cred] (-.25,2) -- node[at start,left] {$L+\epsilon$} ++(3.5,0); } \draw[cred,dashed] (-.25,1.5) -- node[at start,left] {$L$} ++(3.5,0); \onslide<3->{ \draw[fill=cgreen,opacity=.2,draw=none] (1.4,-.25) rectangle ++(.7,3.5); \draw[cgreen] (1.4,-.25) -- node[at start,rotate=90,left] {$a-\delta$} ++(0,3.5); \draw[cgreen] (2.1,-.25) -- node[at start,rotate=90,left] {$a+\delta$} ++(0,3.5); } \draw[cgreen,dashed] (1.75,-.25) -- node[at start,below] {$a$} ++(0,3.5); \draw[cblue,ultra thick] (-.5,2.5) to[out=0,in=-105,looseness=2.2] (3,2); \end{tikzpicture} } \end{minipage}~ \begin{minipage}{.49\textwidth} \pause For every \textcolor{cred}{interval $I_L$ around $L$},\\[1ex]% \pause% find \textcolor{cgreen}{interval $I_a$ around $a$}\\[1ex]% \pause% such that\\[1ex]% if we restrict the domain of $f$ to \textcolor{cgreen}{$I_a$}, then the curve lies in \textcolor{cred}{$I_L$}. \end{minipage} \vspace{10cm} \end{frame}