%% Math 131, Fall, 2003
\documentclass[11pt]{article}
\usepackage{latexsym,amsmath,epic,eepic,pstcol}

\usepackage{math131}

\pagestyle{empty}
\newcommand{\ddx}[1]{\ds\frac{d}{d#1}}

\begin{document}

\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Professor Hwang} \\
Worksheet for Thurssday, November~6
\end{center}

\noindent Calculate the following limits:
  \[
  \lim_{x\to1}\frac{\ln x}{x-1},\qquad
  \lim_{x\to0}\frac{1-\cos x}{x},\qquad
  \lim_{x\to0}\frac{e^x-1-x}{x^2},\qquad
  \lim_{x\to0}\frac{x-\sin x}{x^3}.
  \]

  \vfil

  \[
  \lim_{x\to\infty}\frac{x}{e^x},\qquad
  \lim_{x\to\infty}\frac{x^2}{e^x},\qquad
  \lim_{x\to\infty}\frac{x^n}{e^x} \quad(n>0).
  \]

  \vfil

  \[
  \lim_{x\to\infty}\frac{\ln x}{x},\qquad
  \lim_{x\to\infty}\frac{\ln x}{\sqrt{x}},\qquad
  \lim_{x\to\infty}\frac{\ln x}{x^{0.001}}.
  \]

  \vfil

\noindent We say that ``$g$~grows faster than~$f$'' if
$\lim\limits_{x\to\infty}\frac{f(x)}{g(x)} = 0$. Put the functions
below in increasing order of growth rate.

\[
e^x,\quad \ln(\ln x),\quad x^{100},\quad e^{x^2}, \quad x^{1/2},\quad
\ln x,\quad e^{2x},\quad 1000x^{10},\quad x^x
\]

\eject

\end{document}
