Unformatted text preview: n is O ( n !). (Hint: induction.) (b) Prove without using limits that n 2 is not Ω( n 4 ). (c) [5] Is it true that 2 n is Θ(3 n )? Justify your answer with an appropriate argument that does not use limits. 6. The deﬁnition of the limit of a function f : Z + → R is: lim n →∞ f ( n ) = L ∈ R if and only if for every real number ± > there exists N (which depends on f and ± ) such that, L± < f ( n ) < L + ± for all n ≥ N . (a) [4] Suppose f : Z + → R + ∪ { } (so f takes only nonnegative values). Use the deﬁnition above to prove that if lim n →∞ f ( n ) = L then there exists N such that f ( n ) ≤ L + 1 for all n ≥ N . (b) [6] Suppose f, g : Z + → R + ∪ { } (so f and g take only nonnegative values). Prove that if lim n →∞ f ( n ) /g ( n ) = L , then f ( n ) is O ( g ( n )). Prove further that if L > 0 then f ( n ) is Θ( g ( n ))....
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This note was uploaded on 02/05/2011 for the course MATH 377 taught by Professor Stephenlang during the Spring '11 term at University of Victoria.
 Spring '11
 stephenlang
 Math, Calculus

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