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        riddles >> general problem-solving / chatting / whatever >> Limits of f(x), f'(x), f''(x)?
        
(Message started by: knightfischer on Mar 13^th, 2008, 8:49am)

Title: Limits of f(x), f'(x), f''(x)?
Post by knightfischer on Mar 13^th, 2008, 8:49am

On the GRE Math practice exam there is a question that says if lim (x to inf) of f(x) and f'(x) both exist and are finite, then what can else can we say about lim (x to inf) of f'(x) and/or f''(x)?

The answer is lim (x to inf) f'(x)=0, which is clear. However, another choice is lim (x to inf) f''(x)=0, which is not correct, but I cannot think of a function that meets the criteria of the original question and where lim (x to inf) f''(x) <>0.

Can anyone help?

Title: Re: Limits of f(x), f'(x), f''(x)?
Post by towr on Mar 13^th, 2008, 9:56am

Perhaps if f''(x) alternates around zero at an increased rate. For example take f''(x) = sin(x²)
Although, hopefully there are examples that give a prettier integral.

(for sin(x²) we get http://en.wikipedia.org/wiki/FresnelS It seems to converge, so subtract the limit and you've got your f'(x) -> 0)

Title: Re: Limits of f(x), f'(x), f''(x)?
Post by knightfischer on Mar 13^th, 2008, 1:17pm

Thank you. I did not think of a trig function. Thanks again.

Title: Re: Limits of f(x), f'(x), f''(x)?
Post by rmsgrey on Mar 14^th, 2008, 10:19am

The key point here, of course, is that f''(x) would have to go to 0 if it converges

Unless I'm missing something, the following satisfies the constraints on f and f' while allowing f'' to become unbounded as x increases:

f(x)=x^-3sin(x³)

Some quick approximations give:
f(x) is bounded above by x^-3
f'(x) is bounded above by O(x^-1)
f''(x) is bounded above by O(x)