\documentclass{exam}[12pt]

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\begin{document}
\pagestyle{head}

\firstpageheader{\large Calculus I}{\bf\Large L\'Hopital's Rule}{\large Name:\hspace{1in} }
%\extraheadheight{.7in}
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%\small{\textit{Work all questions completely.  Show all work.}}

\begin{questions}
\large{
\question  Find $\displaystyle \lim_{x\to 2}\frac{x^2-x-2}{x-2}$.
\vspace{1.5in}

\question  Find $\displaystyle \lim_{x\to 0} \frac{\sin x}{x}$.
\vspace{1.2in}

\question Explain why the ``tower of power'' works.  That is, explain why,
if $f$ is higher on the tower than $g$ that
\[
\lim_{x\to\infty}\frac{f(x)}{g(x)} = \infty
\]
and
\[
\lim_{x\to\infty}\frac{g(x)}{f(x)} = 0.
\]
Provide several examples.
\pagebreak

\question Find $\displaystyle \lim_{x\to\infty} 
\left(1+4x\right)^{\frac{3}{x}}$.
\vspace{2.2in}

\question  Find $\displaystyle \lim_{x\to 0^+} \frac{1}{x}-\frac{1}{\sin x}$.
\vspace{2.2in}

\question  Find $\displaystyle \lim_{x\to \infty} 
\left( 1 +\frac{3}{x}\right)^{2x}$.
\vspace{2.2in}


}
\end{questions}
\end{document}