Properties of Asymptotic functions

[keebs]

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...

 

[rachmaninoff]

Think about $e^{(x+1)}=e \cdot e^x$; look at the definitions.

 

[keebs]

Ahhh, ok. Thank you.

 

[Hurkyl]

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.

 

[keebs]

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?

 

[Hurkyl]

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

 

[keebs]

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.