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big ohhh notation :P


Sarahisme
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hey could someone please give me a bit of help with this big O notation bussiness, it makes not much sense to me, (mainly in terms of calucating limit sof indeterminate form, and how the big O's cancel etc.) anyways any help or pointers to a good website that explains the stuff would be great!

 

Thanks Guys and Gals :)

 

Sarah

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If f(x)=O(g(x)) that just means that for large enough x and some constant c, |f(x)|<=c*g(x). You can use this to give bounds on certain limits, like f(x), if f(x)=g(x)+O(h(x)), because we know that lim x->infinity g(x)-h(x) <= lim x->infinity f(x) <= lim x->infinity g(x)+h(x). I don't really know what you mean by "how the big O's cancel," so I can't help you out there.

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What they mean is that there is an error term, on the order of x2 and order of x, in reference to the last line ofc.

 

Since you are taking the limit as x approaches 0, x will become very small, so x2 << 1, and x << 1/3.

 

Thus, they are negligible, and can be tossed away.

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If x < 1, and you're looking at a limit as x -> 0, then x >> x2 >> x3, etc

 

You if you have a term of order x, you can ignore the higher order terms. They are small compared to x. If you have terms that look like x2, you can ignore terms that look like x3, again, because they are small when x -> 0.

 

The O(xn) just means that all the terms are that exponent, n, and the terms are small so they can be ignored. Thus it's not worth the time to actually calculate that term.

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