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Topic: Limits of f(x), f'(x), f''(x)? (Read 374 times) |
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knightfischer
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Limits of f(x), f'(x), f''(x)?
« on: Mar 13th, 2008, 8:49am » |
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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?
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towr
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Re: Limits of f(x), f'(x), f''(x)?
« Reply #1 on: Mar 13th, 2008, 9:56am » |
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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)
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knightfischer
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Re: Limits of f(x), f'(x), f''(x)?
« Reply #2 on: Mar 13th, 2008, 1:17pm » |
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Thank you. I did not think of a trig function. Thanks again.
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rmsgrey
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Re: Limits of f(x), f'(x), f''(x)?
« Reply #3 on: Mar 14th, 2008, 10:19am » |
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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)
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