\frametitle{Precise Definition of Limits}
  Recall the definition of limits:
    Suppose $f(x)$ is defined close to $a$ (but not necessarily $a$ itself).
    We write
      \lim_{x\to a} f(x) = L\\[1ex]
      \text{spoken: ``the limit of $f(x)$, as $x$ approaches $a$, is $L$''}
    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 $a$.
  The intuitive definition of limits is for some purposes too vague:
    \item What means `make $f(x)$ arbitrarily close to $L$' ?
    \item What means `taking $x$ sufficiently close to $a$' ?