Thread: relation between asymptotes and big oh notation

1. relation between asymptotes and big oh notation

What does big oh notation have to do with asymptotes?

2. Re: relation between asymptotes and big oh notation

Originally Posted by Jskid
What does big oh notation have to do with asymptotes?
If g(x) is an asymptote to f(x), what can we say using big-O notation about f and g?

CB

4. Re: relation between asymptotes and big oh notation

Originally Posted by Jskid
Exactly, but it does not work the other way around.

CB

5. Re: relation between asymptotes and big oh notation

Originally Posted by Jskid
Originally Posted by CaptainBlack
Exactly, but it does not work the other way around.
It does not work the other way in the sense that f(x) = O(g(x)) does not imply that g(x) is an asymptote to f(x). However, if g(x) ≠ 0 is an asymptote to f(x), then we have both f(x) = O(g(x)) and g(x) = O(f(x)).

6. Re: relation between asymptotes and big oh notation

Originally Posted by Jskid
Where I'm confused is $\displaystyle x^2$ is $\displaystyle O(x^3)$ but the functions diverge and there's no asymptote?

7. Re: relation between asymptotes and big oh notation

Originally Posted by emakarov
f(x) = O(g(x)) does not imply that g(x) is an asymptote to f(x)
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8. Re: relation between asymptotes and big oh notation

Originally Posted by emakarov
It does not work the other way in the sense that f(x) = O(g(x)) does not imply that g(x) is an asymptote to f(x).
Is it possible that f(x) is an asymptote to g(x) but g(x) is not an asymptote to f(x). If yes I still don't see how $\displaystyle x^3$ is an asymptote to $\displaystyle x^2$ or $\displaystyle x^2$ is an asymptote to $\displaystyle x^3$

9. Re: relation between asymptotes and big oh notation

According to Wikipedia, asymptote is a straight line. Therefore, neither $\displaystyle x^2$ nor $\displaystyle x^3$ can be an asymptote. Yes, it is possible that f(x) is an asymptote to g(x) but g(x) is not an asymptote to f(x) because (the graph of) f(x) is a straight line and g(x) is not. But the fact that f(x) = O(g(x)) does not imply that either f(x) or g(x) is an asymptote to anything or that that either f(x) or g(x) has an asymptote.

10. Re: relation between asymptotes and big oh notation

Originally Posted by emakarov
According to Wikipedia, asymptote is a straight line. Therefore, neither $\displaystyle x^2$ nor $\displaystyle x^3$ can be an asymptote. Yes, it is possible that f(x) is an asymptote to g(x) but g(x) is not an asymptote to f(x) because (the graph of) f(x) is a straight line and g(x) is not. But the fact that f(x) = O(g(x)) does not imply that either f(x) or g(x) is an asymptote to anything or that that either f(x) or g(x) has an asymptote.
Sorry but why is f(x) a straight line?

11. Re: relation between asymptotes and big oh notation

An asymptote is a straight line by definition.

Edit: It may happen that f(x) is an asymptote to g(x) for some f(x) and g(x). This implies that f(x) is a straight line. This does not imply that g(x) is a straight line; therefore, this does not imply that g(x) is an asymptote to anything.

12. Re: relation between asymptotes and big oh notation

Originally Posted by emakarov
An asymptote is a straight line by definition.
Oh I think I'm getting it. In f(x) = O(g(x)) f(x) is not the family of functions that are greater than or equal to O(g(x)), it's the upper bound to it?

13. Re: relation between asymptotes and big oh notation

Originally Posted by Jskid
In f(x) = O(g(x)) f(x) is not the family of functions that are greater than or equal to O(g(x))?
Well, f(x) is not a family of functions; it's one function. You may consider O(g(x)) as a family, but not f(x). Second, why would you think that f(x) = O(g(x)) is somehow related to f(x) greater than or equal to g(x)? If anything, the first, very rough, idea behind f(x) = O(g(x)) is that f(x) <= g(x). (In reality, f(x) <= C * g(x) for some constant C, and this inequality has to hold only eventually.)

14. Re: relation between asymptotes and big oh notation

Here f(n) is O(g(n))

but I see no asymptote?

15. Re: relation between asymptotes and big oh notation

Originally Posted by emakarov
But the fact that f(x) = O(g(x)) does not imply that either f(x) or g(x) is an asymptote to anything or that that either f(x) or g(x) has an asymptote.
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