\begin{exercice}
Soit $f$ la fonction définie sur $\mathbb{R}^*$ par :
\[f(x)=\dfrac{\lfloor x\rfloor}{x^2}\]
représentée dans le repère ci-dessous avec deux autres courbes d'équations $y=\dfrac{1}{x}$ et $y=\dfrac{x-1}{x^2}$.
\begin{center}
\psset{xunit=0.9cm,yunit=0.9cm,algebraic=true}
\begin{pspicture*}(-4.3,-2.2)(4.3,2.4)
\psgrid[subgriddiv=1,linewidth=0.5pt,gridcolor=A3,subgridcolor=A3,gridlabels=0pt](0,0)(-6,-3)(6,3)
\psaxes[linewidth=0.8pt,Dx=1,Dy=1,ticksize=-2pt]{->}(0,0)(-4.3,-2.2)(4.3,2.4)
\psplot[linecolor=blue,plotpoints=5000,linewidth=0.8pt]{-5.1}{-0.001}{1/x}
\psplot[linecolor=blue,plotpoints=5000,linewidth=0.8pt]{0.001}{5.1}{1/x}
\psplot[linecolor=green!60!black,plotpoints=5000,linewidth=0.8pt]{-5.1}{-0.001}{(x-1)/x^2}
\psplot[linecolor=green!60!black,plotpoints=5000,linewidth=0.8pt]{0.001}{5.1}{(x-1)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{-5.1}{-5.001}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{-5}{-4.001}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{-4}{-3.001}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{-3}{-2.001}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{-2}{-1.001}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{-1}{-0.001}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{0.001}{0.999}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{1}{1.999}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{2}{2.999}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{3}{3.999}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{4}{4.999}{floor(x)/x^2}
\psplot[linecolor=red,plotpoints=500,linewidth=0.8pt]{5}{5.1}{floor(x)/x^2}


\rput[0](-4.001,-0.312){\textcolor{Blanc}{$\scriptstyle\bullet$}}
\rput[0](-3.001,-0.444){\textcolor{Blanc}{$\scriptstyle\bullet$}}
\rput[0](-2.001,-0.749){\textcolor{Blanc}{$\scriptstyle\bullet$}}
\rput[0](-1.001,-1.996){\textcolor{Blanc}{$\scriptstyle\bullet$}}
\rput[0](0.999,0){\textcolor{Blanc}{$\scriptstyle\bullet$}}
\rput[0](1.999,0.25){\textcolor{Blanc}{$\scriptstyle\bullet$}}
\rput[0](2.999,0.222){\textcolor{Blanc}{$\scriptstyle\bullet$}}
\rput[0](3.999,0.188){\textcolor{Blanc}{$\scriptstyle\bullet$}}

\rput[0](-4.001,-0.312){\textcolor{red}{$\scriptstyle\circ$}}
\rput[0](-3.001,-0.444){\textcolor{red}{$\scriptstyle\circ$}}
\rput[0](-2.001,-0.749){\textcolor{red}{$\scriptstyle\circ$}}
\rput[0](-1.001,-1.996){\textcolor{red}{$\scriptstyle\circ$}}
\rput[0](0.999,0){\textcolor{red}{$\scriptstyle\circ$}}
\rput[0](1.999,0.25){\textcolor{red}{$\scriptstyle\circ$}}
\rput[0](2.999,0.222){\textcolor{red}{$\scriptstyle\circ$}}
\rput[0](3.999,0.188){\textcolor{red}{$\scriptstyle\circ$}}

\rput[0](-4,-0.25){\textcolor{red}{$\scriptstyle\bullet$}}
\rput[0](-3,-0.333){\textcolor{red}{$\scriptstyle\bullet$}}
\rput[0](-2,-0.5){\textcolor{red}{$\scriptstyle\bullet$}}
\rput[0](-1,-1){\textcolor{red}{$\scriptstyle\bullet$}}
\rput[0](1,1){\textcolor{red}{$\scriptstyle\bullet$}}
\rput[0](2,0.5){\textcolor{red}{$\scriptstyle\bullet$}}
\rput[0](3,0.333){\textcolor{red}{$\scriptstyle\bullet$}}
\rput[0](4,0.25){\textcolor{red}{$\scriptstyle\bullet$}}
\end{pspicture*}
\end{center}
\begin{enumerate}
\item Démontrer que, pour tout réel $x\neq0$ ;
\[\dfrac{x-1}{x^2}< f(x)\leqslant \dfrac{1}{x}.\]
\item En déduire la limite de $f$ en $-\infty$ et $+\infty$.
\end{enumerate}
\end{exercice}

