130 5.3 Limit Superior and Limit Inferior of Functions Remark 5.3.3 Let f : D → R and let x0 be a limit point of D. Suppose liminfx →x0 f (x) is a real number. Defne B = {ℓ ∈ R : ∃{xk}⊂ D,xk ≠ x0 for every k,xk → x0, f (xk) → ℓ}. Then B ̸ / f (x)= minB. = 0 and liminfx →x0 Theorem 5.3.7 Let f : D → R and let x0 be a limit point of D. Then liminf f (x)= −∞ x→x0 if and only if there exists a sequence {xk} in D such that {xk} converges to x0, xk ̸ = x0 for every k, and limk →∞ f (xk)= −∞. Theorem 5.3.8 Let f : D → R and let x0 be a limit point of D. Then liminf f (x)= ∞ x→x0 if and only if for any sequence {xk} in D such that {xk} converges to x0, xk ̸ = x0 for every k, it follows that limk →∞ f (xk)= ∞. The latter is equivalent to limx →x0 f (x)= ∞. Theorem 5.3.9 Let f : D → R, let x0 be a limit point of D, and let ℓ be a real number. Then lim f (x)= ℓ x→x0 if and only if limsup f (x)= liminf f (x)= ℓ. x→x0 x→x 0 Proof: Suppose lim f (x)= ℓ. x→x0 Then for every ε > 0, there exists δ > 0 such that ℓ − ε < f (x) <ℓ + ε for all x ∈ B0(x0;δ ) ∩ D. ′ Since this also holds for every 0 < δ < δ , we get ℓ − ε < g(δ ′ ) ≤ ℓ + ε. It follows that ℓ − ε ≤ inf g(δ ′ ) ≤ ℓ + ε. δ ′ >0 Therefore, limsupx →x0 f (x)= ℓ since ε is arbitrary. The proof for the limit inferior is similar. The converse follows directly from (i) of Theorem 5.3.1 and Theorem 5.3.5. □
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