Does the Limit of f'(x) Imply f''(x) Equals Zero?

[terhorst]

#34 on the much-discussed http://ftp.ets.org/pub/gre/Math.pdf" :

Suppose f is a differentiable function with \lim\limits_{x \to \infty }f(x)=K and \lim\limits_{x \to \infty }f'(x)=L for some K,L finite. Which must be true?

1. L=0
2. \lim\limits_{x \to \infty }f''(x)=0
3. K=L
4. f is constant.
5. f' is constant.

Answer is 1. Is this because f might be C^1? Can you give an example of a function where the limit of the first derivative exists but the limit of the second derivative is not zero? Thanks!

 

[nicktacik]

I am a little befuddled by this. If 1. is true, it seems like 2. must also be true.
Let g(x)=f'(x)
We know
\lim_{x\rightarrow\infty} g(x) = K = 0
So it should follow that
\lim_{x\rightarrow\infty} g'(x) = \lim_{x\rightarrow\infty} f''(x) = L = 0

 

[Dick]

If the limit of the second derivative exists then it is zero. But it may not exist - even if the function is C^2. Try sin(x^2)/x^2.