\displaystyle for a part of the document

With the etoolbox and/or enumitem packages:

\documentclass{article}
\usepackage[showframe, nomarginpar]{geometry}
\usepackage{mathtools}
\usepackage{enumitem}
\usepackage{etoolbox} 
\AtBeginEnvironment{enumerate}{\everymath{\displaystyle}}

\begin{document}

\begin{enumerate}%
        \item $\sin \frac{1}{x},\quad x\in (0,1]$
        \item $\frac{1}{\sin x},\quad x\in \left[\frac{\pi}{6},\frac{5\pi}{6}\right]$
        \item $ \frac{x}{\sin x},\quad x\in \left[\frac{\pi}{6},\frac{\pi}{2}\right]$
\end{enumerate}

\end{document} 

If you want this just for one environment, you can use this (without etoolbox) instead:

\begin{enumerate}[before=\everymath{\displaystyle}]

Use \everymath{\displaystyle}. To make it local, put it inside a group:

\documentclass{article}

\begin{document}
Text style: $\frac{a}{b}$

\bgroup
\everymath{\displaystyle}
Display style: $\frac{a}{b}$
\egroup

And back: $\frac{a}{b}$
\end{document}