We sometime become concern about the asymptotic nature of a function specially when we are considering Limit or nature of a function or composite function at Neighbourhood(Nbd) of a point.
We use following definitions :-
Typically we use these notation for x → ∞.
(i) Big -O :- f = O(g) ↔ ∃ (x0 , ϵ) ∈ R2 s.t. ∀x ≥ x0 |f(x)| ≤ |ϵg(x)| .
This g actually represent a asymptotic upper bound of f .
For more stronger bound we have small-o which is described as :-
f(x) = o(g(x)) ↔ ∀ϵ ∈ R ∃ x0 ∈ R s.t. ∀x ≥ x0 |f(x)| ≤ |ϵg(x)| .
(ii) Big - Ω :- f = O(g) ↔ g = Ω(f). Similarly defined ω.
(iii) Big - Θ :- f = Θ(g) ↔ g = Θ(f) ↔ g = O(f) . Similarly defined θ.
We use following definitions :-
Typically we use these notation for x → ∞.
(i) Big -O :- f = O(g) ↔ ∃ (x0 , ϵ) ∈ R2 s.t. ∀x ≥ x0 |f(x)| ≤ |ϵg(x)| .
This g actually represent a asymptotic upper bound of f .
For more stronger bound we have small-o which is described as :-
f(x) = o(g(x)) ↔ ∀ϵ ∈ R ∃ x0 ∈ R s.t. ∀x ≥ x0 |f(x)| ≤ |ϵg(x)| .
(ii) Big - Ω :- f = O(g) ↔ g = Ω(f). Similarly defined ω.
(iii) Big - Θ :- f = Θ(g) ↔ g = Θ(f) ↔ g = O(f) . Similarly defined θ.
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