Properties of Asymptotic functions

  • Thread starter keebs
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  • #1
keebs
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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...
 

Answers and Replies

  • #2
Think about [itex]e^{(x+1)}=e \cdot e^x[/itex]; look at the definitions.
 
  • #3
keebs
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Ahhh, ok. Thank you.
 
Last edited:
  • #4
Hurkyl
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If h(x)~g(x), is h(x+1)~g(x)?

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.
 
  • #5
keebs
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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.

What about with the prime counting function? Is pi(x+1)~x/lnx?
 
  • #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
 
  • #7
keebs
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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

Ah, ok. Because if either one of those is true then it implies that pi(x+1)~x/lnx.
 

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