\begin{frame} \frametitle{Precise Definition of Limits} \precise \pause\bigskip In words: No matter what $\epsilon > 0$ we choose, \begin{itemize} \item[] if the distance of $x$ to $a$ is smaller than $\delta(\epsilon)$ (and $x\ne a$) \item[] then the distance of $f(x)$ to $L$ is smaller than $\epsilon$. \end{itemize} \pause\bigskip We can make $f$ \alert{arbitrarily close} to $L$ by taking $\epsilon$ arbitrarily small. \pause\\[1ex] Then $x$ is \alert{sufficiently close} to $a$ if the distance is $< \delta(\epsilon)$. \vspace{10cm} \end{frame}