# GRE math subject test prep

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1. Jul 20, 2007

### terhorst

#34 on the much-discussed GRE practice test:

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!

Last edited: Jul 20, 2007
2. Jul 20, 2007

### 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
$$\lim_{x\rightarrow\infty} g(x) = K = 0$$
$$\lim_{x\rightarrow\infty} g'(x) = \lim_{x\rightarrow\infty} f''(x) = L = 0$$