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{\bf Math 166: Honors Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

{\bf Quiz 6}\quad {\sl Mar. 4, 1999}

\begin{enumerate}
\problemspace
\item 
  \begin{enumerate}[a)]
  \item Define what the notation $f(x) = o(g(x))$ as $x\to a$ means.

  \answerspace

  \problemspace
  \item Prove (using the definition) that $x^n o(x^m) = o(x^{n+m})$
  as $x\to 0$ for any integers $n$, $m$.

  \answerspace

  \problemspace
  \item Prove (using the definition and part b)) that $x^2+o(x^3) = o(x)$
  as $x\to 0$.

  \answerspace

  \end{enumerate}

\newpage
\item Compute the limits using Taylor polynomial approximations.
  \begin{enumerate}[a)]

  \item
  $\lim_{x\to0} \Big( \frac{1}{x^2} - \frac{1}{\exp(x^2)-1} \Big)$

  \vfil

  \item 
  $\lim_{x\to0} [\cos(x)]^{a/x^2}$ \qquad
  [Hint: write $\cos(x)^{a/x^2} =\exp[a \log(\cos(x))/x^2]$.]


  \end{enumerate}

\end{enumerate}

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