\begin{frame}
  \frametitle{Precise Definition of Limits}
  
  Recall the definition of limits:
  \begin{block}{}
    Suppose $f(x)$ is defined close to $a$ (but not necessarily $a$ itself).
    We write
    \begin{gather*}
      \lim_{x\to a} f(x) = L\\[1ex]
      \text{spoken: ``the limit of $f(x)$, as $x$ approaches $a$, is $L$''}
    \end{gather*}
    if we can make the values of $f(x)$ arbitrarily close to $L$
    by taking $x$ to be sufficiently close to $a$ but not equal to $a$.
  \end{block}
  \pause\bigskip
  
  \begin{alertblock}{}
  The intuitive definition of limits is for some purposes too vague:
  \begin{itemize}
    \pause
    \item What means `make $f(x)$ arbitrarily close to $L$' ?
    \pause
    \item What means `taking $x$ sufficiently close to $a$' ?
  \end{itemize}
  \end{alertblock}
\end{frame}