Little o notation

  • Thread starter zjhok2004
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  • #1
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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 any one show me how to prove this? thanks
 

Answers and Replies

  • #2
STEMucator
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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
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f=o(g) as x -> 0 means lim f/g ->0 as x ->0
 
  • #4
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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.
 
  • #5
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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
 
  • #6
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That is the definition of the little o notation

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

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