Placing a removable discontinuity in the graph of a function using TikZ (adding tick marks)

Question

I would like to graph the function $y = \sqrt{x^{2} - 4}/\sqrt{x - 2}$ on the domain from -5 to 30. This function will be Sy = \sqrt{x + 2}$ restricted to the open interval $(2, \infty)$. So, I would like an open circle at (2,0). Now, a horizontal grid line is passing through it, and the graph of the function is passing through it.

I would like a dashed line drawn down to the x-axis from (2,0), and a tick mark there and below it the label "2." I would also like a tick mark below the x-intercept and below it the label "-2."

\documentclass[10pt]{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
\usetkzobj{all}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\begin{document}

    \begin{tikzpicture}
\begin{axis}[grid=both,
          xmax=41,ymax=6,
          axis lines=middle,
          restrict y to domain=-1.5:6.5,
          enlargelimits,
          axis line style={shorten >=-0.25cm,shorten <=-0.25cm,latex-latex},
          ticklabel style={fill=white}]
\addplot[domain=2:36,mark=none,samples=200] {sqrt(x + 2)} node[fill=white, below]{$y=\frac{x^{2} - 4}{\sqrt{x-2}}$};
\addplot[domain=-2:2,dashed,mark=none,samples=200] {sqrt(x + 2)};
\draw (axis cs:2,2) circle [radius=1.5pt];
\end{axis}
\end{tikzpicture}

\end{document}

Answer 1

To get an extra tick mark at specified coordinates without adding grid lines, use

      extra x ticks={-2,2},
      extra x tick style={grid=none}

To fill the circle with white, use

fill=white

To get a vertical dashed line from (2,2) to (2,0), use

\draw [dashed] (2,2) -- (2,0);

(no need for axis cs: if you're using \pgfplotsset{compat=1.11} or higher).

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\begin{document}

\begin{tikzpicture}
\begin{axis}[grid=both,
          xmax=41,ymax=6,
          axis lines=middle,
          restrict y to domain=-1.5:6.5,
          enlargelimits,
          axis line style={shorten >=-0.25cm,shorten <=-0.25cm,latex-latex},
          ticklabel style={fill=white},
          extra x ticks={-2,2},
          extra x tick style={grid=none}
]
\addplot[domain=2:36,mark=none,samples=200] {sqrt(x + 2)} node[fill=white, below]{$y=\frac{x^{2} - 4}{\sqrt{x-2}}$};
\addplot[domain=-2:2,dashed,mark=none,samples=200] {sqrt(x + 2)};

\draw [densely dashed] (2,2) -- (2,0);
\draw [fill=white] (2,2) circle [radius=1.5pt];
\end{axis}
\end{tikzpicture}

\end{document}