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Question about a function

  1. Jul 10, 2010 #1
    can exist an smooth function with the property

    [tex] y(\infty) =0 [/tex] and [tex] y'(\infty) =1 [/tex] ?

    the inverse case, a function that tends to 1 for big x and whose derivative tends to 0 is quite obvious but this case i am not sure if there will exist
  2. jcsd
  3. Jul 10, 2010 #2


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    I suppose that you mean
    [tex]\lim_{x \to \infty} y(x) = 0[/tex] and [tex]\lim_{x \to \infty} y'(x) = 1[/tex] ?

    Actually, I think that for a smooth function to have a limit at infinity, the derivative should have limit 0 (at least that's what my intuition tells me: for the function to have a limit at infinity, it should become progressively more flat, so it doesn't run away from its limit value).

    I have some other work now, but I will try to prove that rigorously later (if you want, give it a try yourself).
  4. Jul 10, 2010 #3
    It follows easily from MVT.
  5. Jul 10, 2010 #4


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    That's what I figured, but I got caught up in epsilons and deltas on the back of my scrap piece of paper.
    After finishing my Saturday's to-do list I will take a completely blank paper of normal size and try again :)
  6. Jul 10, 2010 #5
    On the second thought, derivative doesn't have to go to zero, unfortunetely. Consider [tex]\frac{\sin (x^2)}{x}[/tex]. It clearly tends to zero, yet the derivative oscilates. Still, the derivative cannot tend to a nonzero number, and this follows from MVT for sure :wink:
    Sorry for the mistake.

    Edit: derivative of [tex]\frac{\sin (x^2)}{\sqrt{x}}[/tex] oscilates unboundedly, while the function goes to zero.
    Last edited: Jul 10, 2010
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