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# Analysis Question Watch

1. I have no clue what O(x) is in this, I'm sure it has something to do with limiting behaviour but I'm not too sure... Thanks for any help!

Show that

2. (Original post by Blue7195)
I have no clue what O(x) is in this, I'm sure it has something to do with limiting behaviour but I'm not too sure... Thanks for any help!

Show that

Big O notation describes the limiting behaviour of a function

Was there more to the question
3. (Original post by Blue7195)
I have no clue what O(x) is in this, I'm sure it has something to do with limiting behaviour but I'm not too sure... Thanks for any help!

Show that

it means it is of the Order x

as x gets very small x3 is very small so this behaves like x

if you have a graphic package plot the cubic and the line y=x and look at the two close to zero
4. (Original post by TenOfThem)
Big O notation describes the limiting behaviour of a function

Was there more to the question
Nope! It's just a show that, question.
5. (Original post by TeeEm)
it means it is of the Order x

as x gets very small x3 is very small so this behaves like x

if you have a graphic package plot the cubic and the line y=x and look at the two close to zero
Thankyou! But it is a show that question, how would you show x^3 + x --> O(x)? Or is just explaining what you said enough?
6. (Original post by Blue7195)
Nope! It's just a show that, question.
Ok

As x gets very small x^3 becomes irrelevant when compared to x

So as x tends to zero the function behaves as x

Edit ... As indeed teeem said

I was fooled as I expected it to be tending to infinity and mis read
7. (Original post by TeeEm)
it means it is of the Order x

as x gets very small x3 is very small so this behaves like x

if you have a graphic package plot the cubic and the line y=x and look at the two close to zero
...
8. (Original post by Blue7195)
Thankyou! But it is a show that question, how would you show x^3 + x --> O(x)? Or is just explaining what you said enough?
I do not do pure maths, so I do not know the level/rigour that you are expected to use.
To me that suffices for the level/type of maths I do.
I am sure some purists will look at this question and if they can add more rigor I am sure they will.
9. (Original post by TeeEm)
I do not do pure maths, so I do not know the level/rigour that you are expected to use.
To me that suffices for the level/type of maths I do.
I am sure some purists will look at this question and if they can add more rigor I am sure they will.
Thankyou! I've never seen a question like it before, it's Uni level, not in the notes anywhere so maybe it was old content they threw out but just wanted to make sure! :P
10. (Original post by Blue7195)
I have no clue what O(x) is in this, I'm sure it has something to do with limiting behaviour but I'm not too sure... Thanks for any help!

Show that

To show formally that as means that you have to show that:

That's the definition of the original statement in terms of something you can calculate. So you need to show that:

is some +ve constant (which is pretty easy).

In general, in order to show that X is true, you need to work from the definition of X.
11. (Original post by atsruser)
To show formally that as means that you have to show that:

That's the definition of the original statement in terms of something you can calculate. So you need to show that:

is some +ve constant (which is pretty easy).

In general, in order to show that X is true, you need to work from the definition of X.
That makes sense, thanks a lot!
12. (Original post by atsruser)
To show formally that as means that you have to show that:

Sorry, but this isn't correct.

There are two issues.

Firstly, the limit doesn't have to exist: e.g. sin(1/x) is O(1) as x->0.

Secondly, even if the limit does exist, it doesn't have to be > 0. e.g. x^2 is O(x) as x->0 (*)

Correct definition is more like: and s.t. for .

Edit: (*) If it helps, informally f(x) = O(g(x)) means f(x) is no larger than g(x). There is also which means f(x) is roughly "the same size" as g(x). In this case if the limit exists it must be > 0. This may be what you were thinking of.
13. (Original post by DFranklin)
Sorry, but this isn't correct.

There are two issues.

Firstly, the limit doesn't have to exist: e.g. sin(1/x) is O(1) as x->0.

Correct definition is more like: and s.t. for .
This seems to be correct - I was trying to express the concept "bounded by g(x)" in some sense and didn't get it quite right.

Edit: (*) If it helps, informally f(x) = O(g(x)) means f(x) is no larger than g(x). There is also which means f(x) is roughly "the same size" as g(x). In this case if the limit exists it must be > 0. This may be what you were thinking of.
Possibly. I was recalling it incorrectly and was too lazy to check. I'd have to go and read up on this stuff properly to make a more sensible response but I don't have the time at the moment.
14. (Original post by atsruser)
Possibly. I was recalling it incorrectly and was too lazy to check. I'd have to go and read up on this stuff properly to make a more sensible response but I don't have the time at the moment.
I did the same thing on another thread today

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