Concept of designing asymptotes

Question

I just read this about how to make asymptotes, but could someone teach me the same method if I want to draw for dfrac{x^2+0.5x+1.5}{x+3} with their asymptotes? Also teach me about the concept of this.

muzimuzhi Z · Accepted Answer

Get the functions of asymptotes: open https://www.wolframalpha.com/, type in  asymptotes (x^2 + 0.5 * x + 1.5)/(x + 3) followed by Enter. The computation results will tell you the two asymptotes are y = x - 2.5 and y = -3.
Draw the image of (x^2 + 0.5 * x + 1.5)/(x + 3), as well as its two asymptotes:

% based on the example given in https://tex.stackexchange.com/a/291629
documentclass[tikz,border=5pt]{standalone}
usepackage{amsmath}
usepackage{pgfplots}
pgfplotsset{compat=1.17}

begin{document}
begin{tikzpicture}
  begin{axis}[
    axis lines = center,
    xlabel = $x$,
    ylabel = $y$,
    xmax = 5,
    xmin = -15,
    ymax = 5,
    ymin = -15,
    domain=-15:5
  ]
    % Image of function y = (x^2 + 0.5 * x + 1.5)/(x + 3)
    addplot[
      restrict y to domain = -15:10,
      samples = 100,
    ] {(x^2 + 0.5 * x + 1.5)/(x + 3)};
    
    % Oblique asymptote at y = x - 2.5
    addplot[dashed] {x - 2.5};
    % Vertical asymptote at x = -3
    addplot[dashed] (-3, x);
  end{axis}
end{tikzpicture}
end{document}

In general,

to draw an oblique asymptote at y = f(x), use addplot[dashed] {f(x)} or (x, f(x)), and
to draw a vertical asymptote at x = c (c is a constant), use addplot[dashed] (c, x).

cis · Answer

documentclass[tikz,border=5pt]{standalone}
usepackage{amsmath}
usepackage{pgfplots}
pgfplotsset{compat=newest}

begin{document}
begin{tikzpicture}[font=footnotesize]
begin{axis}[
axis lines = center,
xlabel=$x$,
ylabel=$y$,
axis line style = {-latex},
xlabel style={anchor=north},
ylabel style={anchor=east},
%xmin=-10,      xmax=7,
ymin=-25,     ymax=25,
ytick={-25,-20,...,25},
restrict y to domain = -30:30,
domain=-10:10, 
enlarge x limits={abs=1.5},
enlarge y limits={abs=3.5},
%clip=false, 
]
addplot[samples=111,  black,]{(x^2 + 0.5*x + 1.5)/(x + 3)} 
       node[above, xshift=-12mm]{$f(x)=dfrac{x^2+0.5x+1.5}{x+3}$};
addplot[dashed] {0.5*x-1}
      node[below=1mm, pos=0.1]{$a(x)=x-2.5$};
addplot[dashdotted, domain=-25:29] ({-3},{x})
      node[left, pos=0.9]{$x_p=-3$};;
end{axis}
end{tikzpicture}
end{document}

Concept of designing asymptotes

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