130 5.3 Limit Superior and Limit Inferior of Functions Remark 5.3.3 Let f : D→Rand let x0 be a limit point of D. Suppose liminfx→x0 f (x) is a real number. Define B={ℓ ∈R: ∃{xk} ⊂D,xk̸ =x0 for every k,xk →x0, f (xk) →ℓ}. Then B̸=0/ and liminfx→x0 f (x) =minB. Theorem 5.3.7 Let f : D→Rand let x0 be a limit point of D. Then liminf x→x0 f (x) =−∞ if and only if there exists a sequence {xk}inDsuch that {xk}converges tox0, xk̸ =x0 for everyk, and limk→∞ f (xk) =−∞. Theorem 5.3.8 Let f : D→Rand let x0 be a limit point of D. Then liminf x→x0 f (x) =∞ if and only if for any sequence {xk} in Dsuch that {xk} converges to x0, xk̸ =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 x→x0 f (x) =ℓ if and only if limsup x→x0 f (x) =liminf x→x0 f (x) =ℓ. Proof: Suppose lim x→x0 f (x) =ℓ. Then for everyε >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 δ′>0 g(δ′) ≤ℓ+ε. 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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