Trying to plot the equation of a circle of radius $5$ in LaTex

Firstly, in you example, sqrt{...} should be replaced with sqrt(...).

Next, My answer based on the example in pgfplots manual, sec. 4.23, shows that to draw a circle with polar coordinate system, you get smoother plot with fewer samples.

documentclass{article}
usepackage{pgfplots}
pgfplotsset{compat = newest}

begin{document}

Here is a circle.
newline 
begin{tikzpicture}
  begin{axis}[
    axis equal
  ]
    addplot[
      domain = 0:2*pi,
      samples = 40,
      smooth,
      data cs=polar,
      thick,
      blue,
    ] (deg(x), 5);
  end{axis}
end{tikzpicture}
end{document}

Though I am new to pgfplots but maybe this answer can point in the right direction --as copied from--

Plot circles on axis in pgfplots

Need to use axis equal=true to ensure that the scale used on the x-axis is the same as that on the y-axis. Otherwise, the circle will look like an ellipse.

I shifted the placement of the legend by setting legend style -- using --https://tex.stackexchange.com/a/227101/197451

tikzset{My Style/.style={samples=100, thick}}%<----------------preamble
    begin{tikzpicture}
    begin{axis}[
    axis lines = middle,
    xlabel = $x$,
    ylabel = {$y$},
    axis equal=true,
    legend style={at={(0.5,-0.1)},anchor=north,
%       cells={anchor=west},
%       legend pos=north west,
%       legend image post style={yshift=1cm}
    },
    ]
    
%   addplot [domain=-4:4, My Style, red] {2*x+1};
%   addlegendentry{$y=2x+1$}
    
    addplot [domain=-3:3, My Style, blue] {sqrt(9-x^2)};
    addplot [domain=-3:3, My Style, blue] {-sqrt(9-x^2)};
    addlegendentry{$x^2+y^2=9$}
    
    end{axis}
    end{tikzpicture}