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Properties of Asymptotic functions

  1. Jul 10, 2005 #1
    I have a few questions about asymptotic functions, and was wondering if anyone could help...

    If h(x)~g(x), is h(x+1)~g(x)?
    And, if h(x)~g(x), is h(x)h(x+1)~g(x)g(x+1)?

    Thanks in advance for any help...
     
  2. jcsd
  3. Jul 11, 2005 #2
    Think about [itex]e^{(x+1)}=e \cdot e^x[/itex]; look at the definitions.
     
  4. Jul 11, 2005 #3
    Ahhh, ok. Thank you.
     
    Last edited: Jul 11, 2005
  5. Jul 11, 2005 #4

    Hurkyl

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    I'm not sure that implication holds in general... I can imagine failure can occur if the functions grow sufficiently fast, or if they can do other odd things, like zig-zag back and forth.
     
  6. Jul 11, 2005 #5
    What about with the prime counting function? Is pi(x+1)~x/lnx?
     
  7. Jul 11, 2005 #6

    Hurkyl

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    I think it might be easier to first decide if pi(x) ~ pi(x + 1), or if (x - 1) / ln (x - 1) ~ x / ln x
     
  8. Jul 11, 2005 #7
    Ah, ok. Because if either one of those is true then it implies that pi(x+1)~x/lnx.
     
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