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


{\bf Question}

Show that if $f(x) \sim g(x)$ as $x \to \infty$, then
$f(x)=\{1+o(1)\}g(X)$ as $x \to \infty$.

\vspace{.5in}

{\bf Answer}

$\lim_{x\to \infty} \ds\frac{f(x)}{g(x)}=1 \Leftrightarrow f \sim
g,\ x \to \infty$

Thus $\ds \lim_{x \to \infty} \left(\frac{f(x)}{g(x)}-1\right)=0$

$\ds \Rightarrow \frac{f(x)}{g(x)}-1=o(1) \Rightarrow
\un{f(x)=[1+o(1)]g(x)}$



\end{document}