# How do you solve  e^(-x)-x+2=0?

Jun 21, 2016

2.1200, nearly.

#### Explanation:

If LHS is $f \left(x\right) , f \left(2\right) = 0.135 \ldots > 0 \mathmr{and} f \left(3\right) = - 0.950 \ldots < 0$.

So, the root in (2, 3)

Use the method of successive approximation, from the difference

equation

${x}_{n} = 2 + {e}^{- \left({x}_{n - 1}\right)} , n = 1 , 2 , 3 , . .$,

with the starter guess-value ${x}_{0} = 2$,

we obtain the sequence of approximations

2.21..., 2.118..., 2.1202.., 2.1200.., 2.1200..

$f \left(2.12\right) = O \left({10}^{- 5}\right)$.

f(x) is a decreasing function and, therefore, the root is unique.

Important note:
Despite that we get a good approximation to the solution of the given equation, this sequence converges to the solution of the difference equation and not the solution of the given equation, in mathematical exactitude. .