Proving Little o Notation: f(x) = o(g(x)) as x → 0

[zjhok2004]

Homework Statement

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 anyone show me how to prove this? thanks

 

[STEMucator]

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?

 

[zjhok2004]

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

 

[STEMucator]

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

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

Then x²/x³ = 1/x → ±∞ as x → 0.

 

[zjhok2004]

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

Then x²/x³ = 1/x → ±∞ as x → 0.

That is the definition of the little o notation

 

[STEMucator]

That is the definition of the little o notation

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