Add 1 Then Reciprocate and Fibonacci Numbers

[Thomas Klemm]

You can calculate the limes as the fixed point of the iteration:
\[
x=\frac{1}{x+1}
\]
This results in:
\[
x^2+x=1
\]
There are two solutions. One of them is attractive: \(\frac{\sqrt{5}-1}{2}\approx 0.61803\).

Cheers
Thomas