Presentation of an exam using Tikz
With the help of the geoemtry package you can adjust the margin sized. In the following MWE, I have also added marginpar to write the text in the right margins. Lastly, I have also modified the width of the grid to automatically fit into the textblock.
\documentclass [a4paper,10pt]{report}
%%%% PACKAGE %%%%%%%%%%%%%%%%%%%
%\usepackage{fullpage}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{fancyhdr}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathrsfs}
\usepackage{hyperref}
\usepackage{enumitem}
\usepackage{marginnote}
\usepackage{calc}
\usepackage[left=2cm, right=4.5cm, marginparwidth=3.5cm]{geometry}
\begin{document}
\subsection*{Exercise 1}
\marginpar{Do not write in this column}
\begin{enumerate}
\item Let $f:\mathbb R\to \mathbb R$ the function defined by $f(x)=\frac{\sin(x)}{x}$. Prove it's continuous on $(0,1]$.\\
\begin{tikz}[color=gray!60]
\draw (0,0) grid[step=0.4] (\textwidth-\leftmargin,2.4);
\end{tikz}
\ \\
\item Let $g:\mathbb R\to \mathbb R$ the function defined by $g(x)=\frac{\sin(x)}{x^2}$. Prove it's continuous on $(0,1]$.\\
\begin{tikz}[color=gray!60]
\draw (0,0) grid[step=0.4] (\textwidth-\leftmargin,2.4);
\end{tikz}
\end{enumerate}
\end{document}
\documentclass [a4paper,10pt]{report}
%%%% PACKAGE %%%%%%%%%%%%%%%%%%%
%\usepackage{fullpage}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{fancyhdr}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathrsfs}
\usepackage{hyperref}
\usepackage{enumitem}
\usepackage{marginnote}
\usepackage{calc}
\usepackage[left=2cm-\leftmargin, right=4.5cm, marginparwidth=3.5cm]{geometry}
\begin{document}
\subsection*{Exercise 1}
\begin{enumerate}
\item Let $f:\mathbb R\to \mathbb R$ the function defined by $f(x)=\frac{\sin(x)}{x}$. Prove it's continuous on $(0,1]$.\\
\marginpar{\color{blue} \centering Do not write in this column \begin{tikzpicture}\draw (0,0) -- (\marginparwidth,2); \draw (\marginparwidth,0) -- (0,2); \end{tikzpicture}}
\begin{tikz}[color=gray!60]
\draw (0,0) grid[step=0.4] (\textwidth-\leftmargin,2.4);
\end{tikz}
\ \\
\item Let $g:\mathbb R\to \mathbb R$ the function defined by $g(x)=\frac{\sin(x)}{x^2}$. Prove it's continuous on $(0,1]$.\\
\begin{tikz}[color=gray!60]
\draw (0,0) grid[step=0.4] (\textwidth-\leftmargin,2.4);
\end{tikz}
\end{enumerate}
\end{document}
The following provides code that will fill the horizontally available space with a grid. Horizontally only multiples of the grid-step will be filled. You can specify the line width, the grid step, the height and color:
\documentclass [a4paper,10pt]{report}
%%%% PACKAGE %%%%%%%%%%%%%%%%%%%
\usepackage{fullpage}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{fancyhdr}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathrsfs}
\usepackage{hyperref}
\usepackage{enumitem}
\pgfkeys
{%
/surb/.cd,
,step/.initial=0.4cm
,height/.initial=2.4cm
,line width/.initial=0.4pt
,color/.initial=gray!60
}
\newcommand\surbval[1]{\pgfkeysvalueof{/surb/#1}}
\newcommand\gridfill[1][]
{%
\begingroup
\leavevmode
\pgfkeys{/surb/.cd,#1}%
\null
\leaders\hbox
{%
\tikz\draw[color=\surbval{color}, line width=\surbval{line width}]
(0,0) grid [step=\surbval{step}]
(\surbval{step},\surbval{height})%
(0,0) rectangle (\surbval{step},\surbval{height})
;%
\kern-\surbval{line width}\relax
}%
\hfill
\kern\surbval{line width}\relax
\endgroup
}
\usepackage{showframe} % just to show that it'll not create overfull lines
\begin{document}
\subsection*{Exercise 1}
\begin{enumerate}
\item Let $f:\mathbb R\to \mathbb R$ the function defined by
$f(x)=\frac{\sin(x)}{x}$. Prove it's continuous on $(0,1]$.\\
\gridfill
\item Let $g:\mathbb R\to \mathbb R$ the function defined by
$g(x)=\frac{\sin(x)}{x^2}$. Prove it's continuous on $(0,1]$.\\
\gridfill[height=4cm]
\end{enumerate}
\end{document}
EDIT: Because someone mentioned this to me in an otherwise unrelated chat.
The following adds another key to \gridfill, namely height target, which will set the overall used height just like the height key, but the actual grid will only be drawn in multiples of step, so that the remaining space will be blank (padding to the top). It will be used if height target is bigger than 0pt, and has to be a valid TeX length (so needs a valid unit).
\documentclass [a4paper,10pt]{report}
%%%% PACKAGE %%%%%%%%%%%%%%%%%%%
\usepackage{fullpage}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage{fancyhdr}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathrsfs}
\usepackage{hyperref}
\usepackage{enumitem}
\pgfkeys
{%
/surb/.cd,
,step/.initial=0.4cm
,height/.initial=2.4cm
,height target/.initial=-\maxdimen
,line width/.initial=0.4pt
,color/.initial=gray!60
}
\newcommand\surbval[1]{\pgfkeysvalueof{/surb/#1}}
\newcommand*\surbtmp{}
\newcommand\gridfill[1][]
{%
\begingroup
\leavevmode
\pgfkeys{/surb/.cd,#1}%
\null
\leaders\hbox
{%
\ifdim\surbval{height target}>0pt
\pgfmathsetmacro\surbtmp
{int(\surbval{height target}/\surbval{step})*\surbval{step}}%
\pgfkeyssetvalue{/surb/height}{\surbtmp pt}%
\rule{0pt}{\surbval{height target}}%
\fi
\tikz\draw[color=\surbval{color}, line width=\surbval{line width}]
(0,0) grid [step=\surbval{step}]
(\surbval{step},\surbval{height})%
(0,0) rectangle (\surbval{step},\surbval{height})
;%
\kern-\surbval{line width}\relax
}%
\hfill
\kern\surbval{line width}\relax
\endgroup
}
\usepackage{showframe} % just to show that it'll not create overfull lines
\begin{document}
\subsection*{Exercise 1}
\begin{enumerate}
\item Let $f:\mathbb R\to \mathbb R$ the function defined by
$f(x)=\frac{\sin(x)}{x}$. Prove it's continuous on $(0,1]$.\\
\gridfill[height target=4.3cm]
\item Let $g:\mathbb R\to \mathbb R$ the function defined by
$g(x)=\frac{\sin(x)}{x^2}$. Prove it's continuous on $(0,1]$.\\
\gridfill[height=4cm]
\end{enumerate}
\end{document}