Properties of Asymptotic functions

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The discussion revolves around the properties of asymptotic functions, particularly the implications of h(x)~g(x) on h(x+1) and the product h(x)h(x+1)~g(x)g(x+1). Participants express uncertainty about whether these implications hold universally, noting that rapid growth or oscillatory behavior of functions could lead to failures in the assumptions. The prime counting function is also examined, with questions raised about the asymptotic relationship between pi(x) and pi(x+1), as well as the behavior of (x - 1) / ln(x - 1) compared to x / ln x. Ultimately, establishing these relationships is crucial for understanding the implications for pi(x+1)~x/lnx. The conversation highlights the complexities involved in asymptotic analysis.
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...
 
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Think about e^{(x+1)}=e \cdot e^x; look at the definitions.
 
Ahhh, ok. Thank you.
 
Last edited:
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.
 
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?
 
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
 
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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