Let f: R-->R be a differentiable function such that f(0)=f´(0)=0
Demonstrate that if lim f(x)/f´(x) exists, then that limit is equal to 0.
x→0
here take newton raphsons form of soln
x(n+1)=x(n)-h(f(x(n))/f'(x(n)))
now take limit on both sides
lim(x(n+1)-x(n))=-hlim(f(x(n))/f'(x(n))) where x(n) -> 0
as we see the lhs limit is equal to 0 so for rhs either h=0 or the limit is equal to 0
but as we have taken h as some small value which is not equal to zero
therefore we have our rhs limit as zero