\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\newcommand{\un}{\underline}
\newcommand{\undb}{\underbrace}
\parindent=0pt
\begin{document}


{\bf Question}

Determine the order of the following expressions as $z \to
+\infty$, giving suitable implied constants.

\begin{description}
\item[(a)]
$4\pi^2z$

\item[(b)]
$1000z^{\frac{1}{2}}$

\item[(c)]
$\sqrt{z\left(1-\ds\frac{1}{z}\right)}$

\end{description}



\vspace{.5in}

{\bf Answer}

\begin{description}
\item[(a)]
$\ds\lim_{z \to \infty} \frac{|4\pi^2z|}{|z|}=4\pi^2 \Rightarrow
4\pi^2z=O(z),\ z \to +\infty,\ \un{(k \geq 4\pi^2,\ \rm{say})}$

\item[(b)]
$\ds\lim_{z \to +\infty}
\frac{|1000z^{\frac{1}{2}}|}{|z^{\frac{1}{2}}|}=1000$

$\Rightarrow 1000z^{\frac{1}{2}}=O(z^{\frac{1}{2}}),\ z \to
+\infty,\ \un{(k \geq 1000,\ \rm{say})}$

\item[(c)]
$\lim_{z \to +\infty}
\ds\frac{\left|z\sqrt{\left(1-\ds\frac{1}{z}\right)}\right|}{|\sqrt{z}|}=1$

$\Rightarrow
\sqrt{z\left(1-\ds\frac{1}{z}\right)}=O(z^{\frac{1}{2}}),\ z \to
+\infty,\ \un{(k \geq 1,\ \rm{say})}$

\end{description}



\end{document}