绘制序列以显示极限，改进代码

Question

我想生成一个好的图形来显示序列极限的视觉定义。到目前为止，这只是一个想法，但我想对其进行改进，例如：

• 我怎样才能删除 $x$ 和 $y$ 轴上的粗线？我指的是数字。$\ell \pm \epsilon$ 没问题。

• 我怎样才能使 $x$ 轴更长，以获得更大的数字？

谢谢！这是代码，带有序言。

\documentclass[a4paper, 14pt]{extarticle}
\usepackage{geometry}
\geometry{total={173mm, 257mm}, left=23mm, top=23mm, right=23mm, bottom=23mm}
\usepackage{amsmath}
\usepackage[bb=stixtwo]{mathalpha}

\usepackage{concmath}

\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary {datavisualization.formats.functions}
\usetikzlibrary{calc,matrix}
\usepackage{arydshln}

\usepackage{pspicture}
\usepackage{physics}
\usepackage{mathtools}


\usetikzlibrary{fit}
\newcommand{\tikznode}[2]{\relax
\ifmmode%
\tikz[remember picture,baseline=(#1.base),inner sep=0pt] \node (#1) {$#2$};
\else
 \tikz[remember picture,baseline=(#1.base),inner sep=0pt] \node (#1) {#2};%
\fi}
\tikzset{box around/.style={
draw,rounded corners,
inner sep=2pt,outer sep=0pt,
node contents={},fit=#1
},      
}


\begin{center}
\begin{tikzpicture}
\begin{axis}[black, thick, 
 ymin = 0,
 ymax = 3,
 xmin = 0,
 xmax = 19,
 axis x line=center,
 axis y line=left]
\addplot[samples at={0,...,16},only marks,mark size=1, blue]{(2*x+1)/(x+2)};
\end{axis}
 \draw[->, thick, black] (0, 0) -- (6.85, 0)
node[right] {$n$};
\draw[thick, dashed] (0, 3.3) -- (6, 3.3);
\draw[thick, dashed] (0, 4.3) -- (6, 4.3);
 \draw[thick, dotted] (0, 3.8) -- (6, 3.8);
\coordinate (A) at (0, 3.3);
  \draw (A) node[left] {$\ell - \epsilon$} node {};
     \coordinate (B) at (0, 4.3);
  \draw (B) node[left] {$\ell + \epsilon$} node {};
 \draw[->, thick, black] (0, 0) -- (0, 5.7)
node[above] {$a_n$};
\end{tikzpicture}
\end{center}

Answer 1

像这样：

代码（我没有重新审视你的序言）：

\documentclass[a4paper, 14pt]{extarticle}
\usepackage{geometry}
\geometry{total={173mm, 257mm}, left=23mm, top=23mm, right=23mm, bottom=23mm}
\usepackage{amsmath}
\usepackage[bb=stixtwo]{mathalpha}

\usepackage{concmath}

\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary {datavisualization.formats.functions}
\usetikzlibrary{calc,matrix}
\usepackage{arydshln}

\usepackage{pspicture}
\usepackage{physics}
\usepackage{mathtools}


\usetikzlibrary{fit}
\newcommand{\tikznode}[2]{\relax
    \ifmmode%
    \tikz[remember picture,baseline=(#1.base),inner sep=0pt] \node (#1) {$#2$};
    \else
    \tikz[remember picture,baseline=(#1.base),inner sep=0pt] \node (#1) {#2};%
    \fi}
\tikzset{box around/.style={
        draw,rounded corners,
        inner sep=2pt,outer sep=0pt,
        node contents={},fit=#1
    },      
}

\begin{document}
    
    \begin{center}
        \begin{tikzpicture}
            \begin{axis}[black, thick, 
                ymin = 0,
                ymax = 3,
                xmin = 0,
                xmax = 25,
                axis x line=center,
                axis y line=left,
                hide x axis,
                hide y axis]
                \addplot[samples at={0,...,23},only marks,mark size=1, blue]{(2*x+1)/(x+2)};
            \end{axis}
            \draw[->,  black] (0, 0) -- (6.85, 0)
            node[right] {$n$};
            \draw[thick, dashed] (0, 3.3) -- (6.5, 3.3);
            \draw[thick, dashed] (0, 4.3) -- (6.5, 4.3);
            \draw[thick, dotted] (0, 3.8) -- (6.5, 3.8);
            \coordinate (A) at (0, 3.3);
            \draw (A) node[left] {$\ell - \epsilon$} node {};
            \coordinate (B) at (0, 4.3);
            \draw (0,3.8) node[left] () {$\ell$};
            \draw (B) node[left] {$\ell + \epsilon$} node {};
            \draw[->, thick, black] (0, 0) -- (0, 5.7)
            node[above] {$a_n$};
        \end{tikzpicture}
    \end{center}

\end{document}

编辑：更简单的解决方案仅使用tikz。代码：

\documentclass[tikz,border=.5cm]{standalone}

\begin{document}
    
    \begin{tikzpicture}[scale=2]
        \draw[-latex]   (-.5,0) -- (5.1,0) node[right] {$n$};
        \draw[-latex]   (0,-.5) -- (0,3.1) node[above] {$a_n$};
        \node at (-.2,-.2) (o) {$O$};
        \draw[thin,dashed] (5,2.3)--(0,2.3) node[left] () {$\ell+\epsilon$};
        \draw[thick,dashed] (5,2)--(0,2) node[left] () {$\ell$};
        \draw[thin,dashed] (5,1.7)--(0,1.7) node[left] () {$\ell-\epsilon$};
        \foreach \x in {0,...,30}
        \fill[blue] ({\x/6},{(2*\x+1)/(\x+2)}) circle(1pt); 
    \end{tikzpicture}
    
\end{document}

输出：