Newton's Method special case troubles....

Most analyses of Newton's method assume that ,

where a is the solution of the nonlinear equation.

I'd like to investigate what happens in the special case where but

I'd like to show that:

I'm pretty sure I need to use L'H^ospital's rule, but I haven't figured out how to make it work out...I keep getting the numerator to be zero, but I need to to be one.

After this is done, I have to show that in this case, the iteration function

generated by Newton's method is a contraction. Thanks in advance for any tips to get me going on this!