well according to MATLAB's online documentation, determinants are normally computed using an LU decomposition (similar to matt's approach). If greater accuracy is required (such as when dealing with ill-conditioned matrices), the singular value decomposition is used. More details:
MATLAB's...
how about just this simple one with arguably less demands on the idea of "infinity":
\begin{align*}
x &=& 0.999\overline{9} \\
10x &=& 9.999\overline{9} \\
10x-x &=& 9 \\
9x &=& 9 \\
x &=& 1 \\
\end{align}