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\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

\item What means make $f(x)$ arbitrarily close to $L$' ?
\item What means taking $x$ sufficiently close to $a$' ?