Thread: relation between asymptotes and big oh notation

What does big oh notation have to do with asymptotes?

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

Originally Posted by Jskid

Exactly, but it does not work the other way around.

CB

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)).

Originally Posted by Jskid

Where I'm confused is is but the functions diverge and there's no asymptote?

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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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 is an asymptote to or is an asymptote to

According to Wikipedia, asymptote is a straight line. Therefore, neither nor 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.

Originally Posted by emakarov

According to Wikipedia, asymptote is a straight line. Therefore, neither nor 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?

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.

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?

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.)

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


but I see no asymptote?

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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