Properties of Asymptotic functions

  • Context: Graduate 
  • Thread starter Thread starter keebs
  • Start date Start date
  • Tags Tags
    Functions Properties
Click For Summary

Discussion Overview

The discussion revolves around the properties and implications of asymptotic functions, specifically examining relationships between functions and their behavior as they approach limits. Participants explore various scenarios and conditions under which certain asymptotic relationships may hold.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question whether the implication h(x+1)~g(x) follows from h(x)~g(x), suggesting that it may not hold in general due to potential rapid growth or oscillatory behavior of the functions.
  • There is a proposal to consider the prime counting function, specifically whether pi(x+1)~x/lnx, as a case study for asymptotic relationships.
  • One participant suggests that it may be beneficial to first establish whether pi(x)~pi(x+1) or (x-1)/ln(x-1)~x/ln(x) before drawing conclusions about pi(x+1)~x/lnx.

Areas of Agreement / Disagreement

Participants express uncertainty and debate the validity of certain implications regarding asymptotic functions, indicating that multiple competing views remain without a clear consensus.

Contextual Notes

Participants note potential limitations in the implications discussed, particularly concerning the growth rates and behaviors of the functions involved, which may affect the validity of the asymptotic relationships.

keebs
Messages
19
Reaction score
0
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...
 
Mathematics news on Phys.org
Think about [itex]e^{(x+1)}=e \cdot e^x[/itex]; 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.
 

Similar threads

Undergrad Best way to fit three functions
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
4K
MHB Solving g(x)-h(x) <0: Find a, b, c, d
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Is long division for a non-polynomic function possible?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Dfns group action & group, written in terms of the other
  • · Replies 26 ·
Replies
26
Views
2K
MHB How should I interpret a formula without brackets?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad My attempt to understand horizontal transformations of functions
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Comparing ##a^b## and ##b^a##
  • · Replies 19 ·
Replies
19
Views
2K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
2K
MHB The shifting of h in vertex form
  • · Replies 1 ·
Replies
1
Views
2K