Use the bisection method to find the root of:
x = e^-x
Read "The Method" from here.
Bisection method - Wikipedia, the free encyclopedia
Choose and as endpoints for the first iteration
Now find c the midpoint that bisects the interal.
find and
You will find
So now as choose it and to be the new interval as they have opposite signs as so on. The interval will get smaller and converge on the solution.