In another form I presented a proof (post #145)that the lim of 0.9999....=1 using epsilon delta arguments. I was asked to explain each step so I am moving it here for the latex support.

I start with what DanLanglois presents presents in post #139

I do not prove this but it follow from basic arithmetic (by the way numbers carry in addition)....

Quite a while back I had a brief interest in wheather the Laplace transform made numerical sense. I was interested in this for its own sake:

Laplace Transform Convergence
Laplace Transform (Numeric Computation?)

and because if we could numericaly understand the Laplace transform it might tell us more about a dynamic system then a fourier transform.

using the fft to calculate the laplace transform

Memory of my failures in obtaining...

For simplicity let's consider a very simple ODE.

[tex]\dot{x_1}=a x_1^2[/tex]

We can approximate this first order system with a second order ODE as follows:

[tex]\left[ \begin{array}{c}
\dot{x_1} \\
\dot{x_2} \end{array} \right]
=
\left[ \begin{array}{ccc}
0 & 1) \\
0 &
\frac{d f(x_1)}{d x_1}
\end{array} \right]

\left[ \begin{array}{c}
X_1) \\
X_2 \end{array}...