# Little o notation

1. Feb 12, 2013

### zjhok2004

1. The problem statement, all variables and given/known data
Given two functions f and g with derivatives in some interval containing 0, where g is positive. Assume also f(x) = o(g(x)) as x → 0. Prove or disprove each of the following statements:

a)∫f(t) dt = o(∫g(t)dt) as x → 0 (Both integrals goes from 0 to x)
b)derivative of f(x) = o( derivative of g(x)) as x → -

Can any one show me how to prove this? thanks

2. Feb 12, 2013

### Zondrina

Recall, $f = o(g)$ implies :

$\forall k>0, \exists a \space | \space f(x) < kg(x), \forall x>a$ where 'k' and 'a' are arbitrary constants.

That's what you meant by "Assume also f(x) = o(g(x)) as x → 0" right?

3. Feb 12, 2013

### zjhok2004

f=o(g) as x -> 0 means lim f/g ->0 as x ->0

4. Feb 12, 2013

### Zondrina

Not true, what about f = x2, then x2 = o(x3) ( For example ).

Then x2/x3 = 1/x → ±∞ as x → 0.

5. Feb 12, 2013

### zjhok2004

That is the definition of the little o notation

6. Feb 12, 2013

### Zondrina

No, the definition is what I've given you.