107 lines
3.6 KiB
TeX
107 lines
3.6 KiB
TeX
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\title{Limite de suite - Limites}
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\tribe{Terminale Sti2d}
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\date{Novembre 2019}
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%\geometry{left=10mm,right=10mm, top=10mm}
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%\pagestyle{empty}
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\begin{document}
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\section{Limite de suite}
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D'après le précédent chapitre sur les suites, on se rappelle que si on a un nombre réel $q$ strictement positif alors
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\begin{itemize}
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\item Si $q\in \intOO{0}{1}$ alors
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\begin{minipage}{0.5\textwidth}
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\[
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\lim_{n\rightarrow+\infty} q^n = 0
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\]
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\end{minipage}
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\begin{minipage}{0.5\textwidth}
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\begin{tikzpicture}[yscale=1.2, xscale=0.5]
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\tkzInit[xmin=0,xmax=10,xstep=1,
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ymin=0,ymax=1,ystep=.5]
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\tkzGrid
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\tkzAxeXY[up space=0.25,right space=1]
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\global\edef\tkzFctLast{0.6^x}
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\foreach \va in {0, 1, ..., 10}{%
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\tkzDefPointByFct[draw](\va)%
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}
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\end{tikzpicture}
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\end{minipage}
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\item Si $q>1$ alors
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\begin{minipage}{0.5\textwidth}
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\[
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\lim_{n\rightarrow+\infty} q^n = +\infty
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\]
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\end{minipage}
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\begin{minipage}{0.5\textwidth}
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\begin{tikzpicture}[yscale=0.5, xscale=1]
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\tkzInit[xmin=0,xmax=7,xstep=1,
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ymin=0,ymax=20,ystep=5]
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\tkzGrid
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\tkzAxeXY[up space=1,right space=0.5]
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\global\edef\tkzFctLast{1.5^x}
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\foreach \va in {0, 1, ..., 7}{%
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\tkzDefPointByFct[draw](\va)%
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}
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\end{tikzpicture}
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\end{minipage}
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\end{itemize}
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On peut lire la limite d'une suite graphiquement comme vu en exercice.
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\begin{itemize}
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\item Pour la suite $w_n = 3n^3 - 10n^2 + 1$
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\begin{minipage}{0.4\textwidth}
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\[
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\lim_{n\rightarrow+\infty} w_n = +\infty
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\]
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Même si la suite est au début décroissante ce qui nous intéresse c'est son comportement quand $n$ devient grand. Ici $(w_n)$ grandit indéfiniement
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\[
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w_{10} = 2001 \qquad w_{100} = 2900001
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\]
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\end{minipage}
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\hfill
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\begin{minipage}{0.5\textwidth}
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\begin{tikzpicture}[yscale=.3, xscale=2]
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\tkzInit[xmin=0,xmax=4,xstep=1,
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ymin=-20,ymax=50,ystep=5]
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\tkzGrid
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\tkzAxeXY[up space=1,right space=0.2]
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\global\edef\tkzFctLast{3*x^3 - 10*x^2 + 1}
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\foreach \va in {0, 1, ..., 4}{%
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\tkzDefPointByFct[draw](\va)%
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}
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\end{tikzpicture}
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\end{minipage}
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\item Pour la suite $z_n = 1 + 5\times0.5^n$
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\begin{minipage}{0.4\textwidth}
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\[
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\lim_{n\rightarrow+\infty} z_n = 1
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\]
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Dans ce cas on a \textbf{asymptote horizontale} d'équation $y=1$ (en rouge sur le graphique). Plus $n$ est grand plus la valeur de $z_n$ se rapproche de 1.
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\end{minipage}
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\hfill
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\begin{minipage}{0.5\textwidth}
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\begin{tikzpicture}[yscale=.5, xscale=0.8]
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\tkzInit[xmin=0,xmax=10,xstep=1,
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ymin=00,ymax=7,ystep=1]
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\tkzGrid
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\tkzAxeXY[up space=1,right space=0.2]
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\global\edef\tkzFctLast{1 + 5*0.5^x}
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\foreach \va in {0, 1, ..., 10}{%
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\tkzDefPointByFct[draw](\va)%
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}
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\tkzHLine[color=red,style=solid,line width=1.2pt]{1}
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\end{tikzpicture}
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\end{minipage}
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\end{document}
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