$(4)$ Let $\backslash ($ f$($x$)=\backslash$ln $($x$)\backslash ).$

$($a$)$ For which initial values $\backslash ($ x$\_\{0\}\backslash )$ are we guaranteed that Newton's Method will not converge? Justify your answer with algebra.

$($b$)$ By the Error Analysis in sec$.3\mathrm{.}2\mathrm{.}1($specifically on p$.85),$ for which initial values $\backslash ($ x$\_\{0\}\backslash )$ is it possible that Newton's Method converges? Express your answer as an interval $(\backslash ($ a$,$ b $\backslash )).$

$-$ Note: It is of course possible that any value of $\backslash ($ x$\_\{0\}\backslash )$ will give convergence, but the point here is that the Error Analysis allows us to restrict our attention to a particular interval.

$($c$)$ Compare your answers in parts $($a$)$ and $($b$).$ If there is an intersection $($ie$,$ common initial values in both answers$),$ does this contradict anything? Why or why not?