Division on a chip that can only multiply

Your hardware can add, subtract and multiply, but it has no division instruction. You need $1/7$ to 12 significant digits.

Apply Newton's method to a function whose root is $1/a$ and whose update needs no division. Show how the error evolves from step to step, find which starting values converge, and count the steps needed from $x_0 = 0.1$. What happens from $x_0 = 0.3$?