How can you use L'Hopital's rule in the limit?
You can't assume that f(a) = 0, which, because f'(a) = 0, must be true to use it. Otherwise your needed conclusion is trivial. Sorry that I can't help you... but maybe you're on the wrong track here?
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!
How can you use L'Hopital's rule in the limit?
You can't assume that f(a) = 0, which, because f'(a) = 0, must be true to use it. Otherwise your needed conclusion is trivial. Sorry that I can't help you... but maybe you're on the wrong track here?
Well, our professor offered LHopital's rule as a hint for the problem. If you apply it to the limit, it yields:
which looks similar to what I'm looking for, except for the pesky f'(x) in the numerator of the right hand side...
I may have left out the fact that we're assuming a is a solution of f(x)=0.