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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 \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}