Why is it that we can start with a number (say, 1), and press cos x repeatedly, we can get the solution of cos x = x?
e.g. cos (1) = 0.54
cos (0.54) = 0.85
cos (0.85) = 0.65
cos (0.65) = 0.79
cos (0.79) = 0.71
cos (0.71) = 0.76
cos (0.76) = 0.72
until it converges to 0.73
This is called Picard's Method (to find roots in R).
It can also be applied to other functions.
If you start with a positive x value, it'll diverge. But a negative initial value will diverge to zero. You can also change the function to to make it diverge for positive x values.