Introduction to Mathematical Analysis I - Second Edition

93 Theorem 3.6.6 Let f : D → R and let ¯ x be a limit point of D . Then ` = liminf x → ¯ x f ( x ) if and only if the following two conditions hold: (1) For every ε > 0, there exists δ > 0 such that ` − ε < f ( x ) for all x ∈ B 0 ( ¯ x ; δ ) ∩ D ; (2) For every ε > 0 and for every δ > 0, there exists x ∈ B 0 ( ¯ x ; δ ) ∩ D such that f ( x ) < ` + ε . Corollary 3.6.7 Suppose ` = liminf x → ¯ x f ( x ) . Then 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 ) = `. Moreover, if { y k } is a sequence in D that converges to ¯ x , y k 6 = ¯ x for every k , and lim k → ∞ f ( y k ) = ` 0 , then ` 0 ≥ ` . Remark 3.6.8 Let f : D → R and let ¯ x be a limit point of D . Suppose liminf x → ¯ x f ( x ) is a real number. Define B = { ` ∈ R : ∃{ x k } ⊂ D , x k 6 = ¯ x for every k , x k → ¯ x , f ( x k ) → ` } . Then B 6 = /0 and liminf x → ¯ x f ( x ) = min B . Theorem 3.6.9 Let f : D → R and let ¯ x be a limit point of D . Then liminf 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 ) = − ∞ . Theorem 3.6.10 Let f : D → R and let ¯ x be a limit point of D . Then liminf 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 ) = ∞ . Theorem 3.6.11 Let f : D → R and let ¯ x be a limit point of D . Then lim x → ¯ x f ( x ) = ` if and only if limsup x → ¯ x f ( x ) = liminf x → ¯ x f ( x ) = `.