(i)[2]
By drawing a suitable pair of graphs, demonstrate that the equation $\ln x = 2 - x^2$ has only one root.
(ii)[2]
Use calculation to verify that this root lies between $x = 1.3$ and $x = 1.4$.
(iii)[1]
Show that, if a sequence produced by the iterative formula $x_{n+1} = \sqrt{2 - \ln x_n}$ converges, then its limit is the root of the equation in part (i).
(iv)[3]
Use the iterative formula $x_{n+1} = \sqrt{2 - \ln x_n}$ to find the root correct to $2$ decimal places. Record each iterate to $4$ decimal places.