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{\bf Math 165: Honors Calculus I}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

{\bf Quiz 6}\quad {\sl Oct.\ 15, 1998}

\begin{enumerate}
\problemspace
\item Give precise definitions for the following.
    \begin{enumerate}[a)]
    \item   $\lim_{x\to p}f(x) = A$.
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    \item   $f(x)$ is continuous at~$p$.
    \answerspace

    \end{enumerate}

\item Let $f(x) = [x]^2$ where $[x]$ is the greatest integer $\le x$.
Use one-sided limits to show that $\lim_{x\to 2} f(x)$ does not exist.

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\item Suppose $f(x)$ is a function that has the following property:
  \[
  \mbox{if $-0.25 < x < 0.25$ then $1 - |x|^3 < f(x) < 1 + |x|^3$}
  \]
The graph of $f(x)$ would thus lie between the curves $y = 1-|x|^3$ and
$y = 1+|x|^3$, at least for $x$ near $0$, and might look something like
the following:
  \[
  \includegraphics{quiz06.eps}
  \]
Prove, \emph{using the definition of limit}, that $\lim_{x\to0} f(x) = 1$.

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