Drawing a random function to show the definition of the limit

On the provided picture the bounds are a little off, so I changed that. A comment on style, your choice of ultra thick for lines is a little weird, also if your prefer the log function, you can replace ln(x) in % f(x) (sigmoid) however you would have to recompute the bounds, in that regard a more elegant solution is to pass the values of the %bounds nodes to % Lower %Upper %Arrows, take it as homework :).

documentclass{minimal}
usepackage{tikz}
usetikzlibrary{calc, math, shapes.geometric, shapes.misc, arrows.meta}

begin{document}

begin{center}
    begin{tikzpicture}[>=stealth]
        % grid and Axis
        draw[help lines, color=gray!30, dashed] (-3 , -3) grid (5 , 4);
    draw[->,ultra thick] (-3 , 0) -- (5 , 0) node[right]{$x$}; 
    draw[->,ultra thick] (0 , -3) -- (0, 4) node[above]{$y$}; 
    % Bounds and function value
    draw[dashed,color=blue] (3.5, 0) node[below right]
            {$scriptstyle x_0+delta$} -- (3.5 , 3.1) -- (0 , 3.1)
            node[left] {$scriptstyle l+varepsilon$}; %Upper
    draw[dashed,] (3  , 0) node[below = 1.5 mm] 
            {$scriptstyle x_0$} -- (3 , 2.55) -- (0 , 2.55)
            node[left] {$scriptstyle l$}; % l
    draw[dashed,color=blue] (2.5 ,  0) node[below left]
            {$scriptstyle x_0-delta$} -- (2.5 , 2.0) -- (0 , 2.0)
            node[left] {$scriptstyle l-varepsilon$}; % Lower
    draw[ultra thick,red,domain=-1.8 : 2.25 , samples = 20] {} 
            plot ((x + 2 , {exp(x + 1.5)/(2 +  exp(x)}); % f(x)
    % Annotations
    draw [|-|, ultra thick, green ] (2.5 , 0) -- (3.0 , 0) 
        node[midway, below = 2.50mm]{$scriptstyle delta$}; 
    draw [<-> , thin , blue] (0.25 , 2.55) -- (0.25 , 3.1)
        node[midway, right] {$scriptstyle varepsilon$}; % Upper Arrow
    draw [<-> , thin , blue] (0.25 , 2) -- (0.25 , 2.55)
            node[midway, right] {$scriptstyle varepsilon$}; % Lower Arrow     
    end{tikzpicture}
end{center}
end {document}