Simple derivation of exponential approximation

I was watching one of Brian Douglas' videos on control theory (Discrete Control #5) and ran into a simple derivation of an approximation I presented earlier.

Back in April I wrote several post on simple approximations for log, exp, etc. In this post I gave an approximation for the exponential function:

The control theory video arrives at the same approximation as follows.

As I believe I've suggested before here, in a derivation like the one above, where you have mostly equalities and one or two approximations, pay special attention to the approximation steps. The approximation step above uses a first order Taylor approximation in the numerator and denominator.

The plot below shows that the approximation above (the bilinear approximation) is more accurate than doing a single Taylor approximation, approximating exp(x) by 1 + x (linear approximation).

Here's a plot focusing on the error in the bilinear and linear approximations.

The bilinear approximation is hard to tell from 0 in the plot above for x up to 0.5.

The derivation above is simple, but why is the result so good? An explanation in terms of Pade approximation is given here.

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