44/115
\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}