Big-O and little-o

Kent Merryfield

The attached file is a handout I'm going to give to a class of mine this week. It's a junior-level "real analysis" class, but there's nothing in here that isn't calculus.

Feel free to suggest changes or additional problems.

**Posted:** Sun Mar 27, 2005 9:08 pm

liyi

Does the infinitesimal $\sin\sqrt{x+1}-\sin\sqrt{x} \ (x\to\infty)$ has an order?

**Posted:** Sun Mar 27, 2005 9:39 pm

Kent Merryfield

**Posted:** Sun Mar 27, 2005 11:51 pm

liyi

I meant to find $o(\frac{1}{x^\alpha})$ ... It is quite easy for big-Oh

**Posted:** Mon Mar 28, 2005 1:10 am

Kent Merryfield

liyi, one can see that

$\limsup_{x\to\infty}\frac{|\sin(\sqrt{x+1})- \sin(\sqrt{x})|}{\frac1{\sqrt{x}}}=\frac12.$

We can do this by choosing $x$ close to $n^2\pi^2.$

So you can't do any better than $O\left(\frac1{\sqrt{x}}\right).$

**Posted:** Mon Mar 28, 2005 7:24 am

Peter

something must be wrong here, but some stuff I cannot read.

some things get replaced by "Fout!" , which is dutch for "Error!"

do I need to install something specific on word? I've always wondered what those O's were...

**Posted:** Mon Mar 28, 2005 9:49 am

_________________
Boo!

Kent Merryfield

Peter - is the .pdf version any better?

**Posted:** Mon Mar 28, 2005 9:59 am

adidasty · Poincare Conjecture Offline Joined: 20 Feb 2005 Posts: 232

I downloaded the .pdf and it looks good, but it looks like you did the document in Microsoft Word and didn't use $\LaTeX$ . It turns out fine though.

**Posted:** Mon Mar 28, 2005 11:53 am

Peter

yes, that works fine, thanks! Smile

**Posted:** Mon Mar 28, 2005 1:20 pm

_________________
Boo!

liyi

**Posted:** Mon Mar 28, 2005 4:19 pm

FMako

Woah this notation is very neat. It's basically making a function that you don't care about as much, except for it's largest polynomial degree depending on whether you're getting big or small. But what is a lim sup?

**Posted:** Tue Nov 15, 2005 2:42 pm

Peter

lim sup = limes superior. Some functions do not have a real 'limit', like f(x)=sin(x). So we cal lim sup the highest value it takes in the limit. e.g. for sin(x) that's simply 1. Smile

**Posted:** Tue Nov 15, 2005 2:45 pm

_________________
Boo!

Kent Merryfield

**Posted:** Tue Nov 15, 2005 3:44 pm

FMako

In your exact definitions of the "o" functions, you used a backwards E among other characters, what does the definition mean in english or could you define some of your notation or link me to a site with that notation defined?

**Posted:** Wed Nov 16, 2005 5:41 pm

Kent Merryfield

It's a logical quantifier. There are two logical quantifiers:

$\exists,$ which means "there exists." ( $\TeX$ code \exists), and

$\forall,$ which means "for all" or "for every." ( $\TeX$ code \forall.)

**Posted:** Wed Nov 16, 2005 7:47 pm

Aunt Sally · Hodge Conjecture Offline Joined: 27 Feb 2006 Posts: 72

That was great. Kent, you're a good explainer. You should write a book.

One thing that has always bugged me is that in numerical analysis, they often say the error is big-O of h, or something like that. But in numerical analysis it seems that we SHOULD care what the constant is! Because if the constant is huge, then in practice our error may be huge. My feeling when I hear numerical analysts say this is, so what, that still doesn't tell me in practice if my error is small or not.

**Posted:** Wed May 03, 2006 11:34 pm

guile · Yang-Mills Theory Offline Joined: 05 May 2005 Posts: 535

**Posted:** Tue Jun 06, 2006 1:06 am

_________________
Cogito Ergo Sum

mdk

I believe that that symbol is the delta and in the delta epsilon form (yoiu know,when you're rigorously trying to prove limits.)

**Posted:** Tue May 22, 2007 5:05 am

quangpbc

I think some problems about $\LaTeX$ we ought to post on Box Latex help Smile

**Posted:** Tue Sep 04, 2007 10:20 pm

bos1234 · Riemann Hypothesis Offline Joined: 24 Jan 2007 Posts: 449

Thanks prof.

I need to become more familiar with the notations. Other than that its a nice paper.

How come you put iff ? What does iff mean?

**Posted:** Mon Oct 15, 2007 4:08 am