Introduction to Mathematical Analysis I - Second Edition

92 3.6 LIMIT SUPERIOR AND LIMIT INFERIOR OF FUNCTIONS Remark 3.6.3 Let f : D → R and let ¯ x be a limit point of D . Suppose limsup x → ¯ x f ( x ) is a real number. Define A = { ` ∈ R : ∃{ x k } ⊂ D , x k 6 = ¯ x for every k , x k → ¯ x , f ( x k ) → ` } . Then the previous corollary shows that A 6 = /0 and limsup x → ¯ x f ( x ) = max A . Theorem 3.6.4 Let f : D → R and let ¯ x be a limit point of D . Then limsup x → ¯ x f ( x ) = ∞ if and only if there exists a sequence { x k } in D such that { x k } converges to ¯ x , x k 6 = ¯ x for every k , and lim k → ∞ f ( x k ) = ∞ . Proof: Suppose limsup x → ¯ x f ( x ) = ∞ . Then inf δ > 0 g ( δ ) = ∞ , where g is the extended real-valued function defined in ( 3.10 ) . Thus, g ( δ ) = ∞ for every δ > 0. Given k ∈ N , for δ k = 1 k , since g ( δ k ) = sup x ∈ B 0 ( ¯ x ; δ k ) ∩ D f ( x ) = ∞ , there exists x k ∈ B 0 ( ¯ x ; δ k ) ∩ D such that f ( x k ) > k . Therefore, lim k → ∞ f ( x k ) = ∞ . Let us prove the converse. Since lim k → ∞ f ( x k ) = ∞ , for every M ∈ R , there exists K ∈ N such that f ( x k ) ≥ M for every k ≥ K . For any δ > 0, we have x k ∈ B 0 ( ¯ x ; δ ) ∩ D whenever k is sufficiently large. Thus, g ( δ ) = sup x ∈ B 0 ( ¯ x ; δ ) ∩ D f ( x ) ≥ M . This implies g ( δ ) = ∞ , and hence limsup x → ¯ x f ( x ) = ∞ . Theorem 3.6.5 Let f : D → R and let ¯ x be a limit point of D . Then limsup x → ¯ x f ( x ) = − ∞ if and only if for any sequence { x k } in D such that { x k } converges to ¯ x , x k 6 = ¯ x for every k , it follows that lim k → ∞ f ( x k ) = − ∞ . The latter is equivalent to lim x → ¯ x f ( x ) = − ∞ . Following the same arguments, we can prove similar results for inferior limits of functions.