\documentclass[12pt]{article}

\usepackage{enumerate}
\usepackage{calculus}
\setlength{\parindent}{0pt}
\pagestyle{empty}
\everymath={\displaystyle}

\begin{document}

\textbf{Math 165: Honors Calculus I}

\textbf{Assignment 16}\quad \emph{Oct. 10, 1999}

\bigskip
\begin{enumerate}
\item Show that $\lim_{x\to0^+}\sqrt{x} = 0$.\newline
(Hint: $0 < x < \delta \Rightarrow 0 < \sqrt{x} < \sqrt{\delta}$.)

\bigskip
\item Show that $\lim_{x\to0^+} x^{1/n} = 0$ for $n\in\N$.

\bigskip
\item Show that $\lim_{x\to0^-} x^{1/n} = 0$ for $n$ an odd positive integer.

\bigskip
\item Let $f(x) = {|x|\over x}$ for $x \ne 0$. Use one-sided limits to show
that $\lim_{x\to0} f(x)$ does not exist.

\bigskip
\item Let $f(x) = x^2$ and let $p\in\R$.
  \begin{enumerate}[a)]
  \item Show that if $|x - p| < 1$ then $|x| < |p| + 1$.\newline
  (Hint: consider $||x| - |p|| \le |x-p|$.)

  \item Show that if $|x-p| < 1$ then $|x^2 - p^2| < (2|p|+1)|x - p|$.

  \item Show that $\lim_{x\to p}x^2 = p^2$.\newline
  (Hint: try $\delta = \min(1,{\epsilon\over 2|p|+1})$.)
  \end{enumerate}

\end{enumerate}

\end{document}

