Lebl-Basic-Analysis-I · Problem 3.5.8

Continuous Functions

Mathematics · limits-at-infinity

Problem

Extend Lemma 3.5.5 as follows. Suppose S⊂RS \subset \mathbb{R} has a cluster point c∈Rc \in \mathbb{R}, c=∞c = \infty, or c=−∞c = -\infty. Let f:S→Rf : S \to \mathbb{R} be a function and suppose L=∞L = \infty or L=−∞L = -\infty. Show that lim⁡x→cf(x)=Lif and only iflim⁡n→∞f(xn)=L for all sequences {xn}n=1∞ such that lim⁡n→∞xn=c.\lim_{x \to c} f(x) = L \quad \text{if and only if} \quad \lim_{n \to \infty} f(x_n) = L \text{ for all sequences } \{x_n\}_{n=1}^\infty \text{ such that } \lim_{n \to \infty} x_n = c.
Intuitive explanation

This problem extends the sequential characterization of limits to the extended real line, where limits and cluster points can be infinite. The key insight is that the ϵ\epsilon-δ\delta definition of limit still works when you replace finite neighborhoods with appropriate "tail" conditions: xx near ∞\infty means x>Mx > M for large MM, and f(x)→∞f(x) \to \infty means f(x)>Kf(x) > K for large KK. The proof strategy mirrors the finite case—convert between the topological definition (using neighborhoods) and the sequential one (using sequences)—but you must carefully translate what "approaching infinity" means in each direction.

Intuitive explanation

This problem extends the sequential characterization of limits to the extended real line, where limits and cluster points can be infinite. The key insight is that the ϵ\epsilon-δ\delta definition of limit still works when you replace finite neighborhoods with appropriate "tail" conditions: xx near ∞\infty means x>Mx > M for large MM, and f(x)→∞f(x) \to \infty means f(x)>Kf(x) > K for large KK. The proof strategy mirrors the finite case—convert between the topological definition (using neighborhoods) and the sequential one (using sequences)—but you must carefully translate what "approaching infinity" means in each direction.

Related theory

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