[Tex/LaTex] How to draw a spiral that gets arbitrary close to a unit circle

tikz-pgf

I would like to draw something like this:

enter image description here

The circle in the center is not connected to the spiral, but the spiral gets arbitrary close to the circle.

A problem why I can't draw this is that I don't know how to describe this mathematically / formally.

I have found this one:

\documentclass[varwidth=true, border=2pt]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
    \draw [domain=0:30,variable=\t,smooth,samples=100]
        plot ({\t r}: {0.005*\t*\t});
\end{tikzpicture}
\end{document}

but it seems not to get closer to a fixed size circle in the center.

This image looks similar to what I'm looking for, but I only need one spiral.

Background of my question

I'm currently studying geometry and topology. Sadly the professor does not provide a script, so I write one by my own to make studying easier for other students (see repository with source files and compiled pdf). The space that is described by the spiral and the circle is connected when you have a definition that makes use of $\varepsilon$-environments, but not connected when you make use of a definition that demands the existance of a path.

Best Answer

You need a function for the radius that approaches 1. For example, you could use 1+2*exp(-0.1*\t):

\documentclass[border=5pt]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
    \draw [red] (0,0) circle [radius=1];
    \draw [domain=0:50,variable=\t,smooth,samples=500]
        plot ({\t r}: {1+2*exp(-0.1*\t)});
\end{tikzpicture}
\end{document}
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