1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: GRE math subject test prep

  1. Jul 20, 2007 #1
    #34 on the much-discussed http://ftp.ets.org/pub/gre/Math.pdf" [Broken]:

    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 by a moderator: May 3, 2017
  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


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook