\begin{frame} \frametitle{Precise Definition of Limits} \precisebook \bigskip \pause \emph{Geometric interpretation:} \medskip \mpause[1]{ For any \textcolor{cred}{small interval $(L-\epsilon,L+\epsilon)$ around $L$}, }\\ \mpause[2]{ we can find \textcolor{cgreen}{an interval $(a-\delta,a+\delta)$ around $a$} }\\ \mpause[3]{ such that $f$ maps all points in \textcolor{cgreen}{$(a-\delta,a+\delta)$} into \textcolor{cred}{$(L-\epsilon,L+\epsilon)$}. } \bigskip \begin{center} \begin{tikzpicture}[default] \draw[->] (0,0) -- ++(5cm,0); \draw[->] (6cm,0) -- ++(5cm,0); \begin{scope}[cgreen,ultra thick] \node[include] at (2.5,0) {}; \node[anchor=north] at(2.5,-.1) {$a$}; \mpause[2]{ \draw[(-)] (1,0) -- node[at start,below,yshift=-1mm] {$a-\delta$} node[at end,below,yshift=-1mm] {$a+\delta$} ++(3,0); } \end{scope} \begin{scope}[cred,ultra thick,xshift=6cm] \node[include] at (2.5,0) {}; \node[anchor=north] at(2.5,-.1) {$L$}; \mpause[1]{ \draw[(-)] (1,0) -- node[at start,below,yshift=-1mm] {$L-\epsilon$} node[at end,below,yshift=-1mm] {$L+\epsilon$} ++(3,0); } \end{scope} \draw[cblue,->,ultra thick,shorten >= 1mm] (1.5,0) to[out=90,in=110,looseness=.3] node[at start,above,yshift=2mm] {$x$} node[at end,above,yshift=3mm] {$f(x)$} (9.5,0); \end{tikzpicture} \end{center} \vspace{10cm} \end{frame}