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GRE math subject test prep

  1. Jul 20, 2007 #1
    #34 on the much-discussed GRE practice test:

    Suppose [tex]f[/tex] is a differentiable function with [tex]\lim\limits_{x \to \infty }f(x)=K[/tex] and [tex]\lim\limits_{x \to \infty }f'(x)=L[/tex] for some [tex]K,L[/tex] finite. Which must be true?
    1. [tex]L=0[/tex]
    2. [tex]\lim\limits_{x \to \infty }f''(x)=0[/tex]
    3. [tex]K=L[/tex]
    4. [tex]f[/tex] is constant.
    5. [tex]f'[/tex] is constant.

    Answer is 1. Is this because [tex]f[/tex] might be [tex]C^1[/tex]? 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. jcsd
  3. Jul 20, 2007 #2
    I am a little befuddled by this. If 1. is true, it seems like 2. must also be true.
    Let [tex]g(x)=f'(x)[/tex]
    We know
    [tex]\lim_{x\rightarrow\infty} g(x) = K = 0 [/tex]
    So it should follow that
    [tex]\lim_{x\rightarrow\infty} g'(x) = \lim_{x\rightarrow\infty} f''(x) = L = 0 [/tex]
  4. Jul 20, 2007 #3


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    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.
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