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Math and science::Analysis::Tao::09. Continuous functions on R

Limits at infinity (for continuous function)

Formulations of the limit \( \lim_{x \to x_0; x \in X \cap E}f(x) \) for a function \( f : X \to \mathbb{R} \), where \( E \subseteq X \subseteq \mathbb{R} \), so far have covered the case where \( x \to x_0 \) where \( x_0 \) is a real number. Below, the idea is extended to describe what it means for limits of \( f \) when \( x_0 \) equals \( +\infty \) or \( -\infty \).

Infinite adherent points

Let \( X \subseteq \mathbb{R} \).

We say that \( +\infty \) is adherent to \( X \) iff for every \( M \in X \) there exists an \( x \in X \) such that \( x > M \).

We say that \( -\infty \) is adherent to \( X \) iff for every \( M \in X \) there exists an \( x \in X \) such that \( x < M \).

In other words, \( +\infty \) is adherent to \( X \) iff \( X \) has no upper bound, or equivalently, \( \sup(X) = +\infty \). Similarly, \( -\infty \) is adherent to \( X \) iff \( X \) has no lower bound, or equivalently, \( inf(X) = -\infty \).

So a set is bounded iff \( +\infty \) and \( -\infty \) are not adherent points.

Limits at infinity

Let \( X \subseteq \mathbb{R} \) with \( +\infty \) being an adherent point, and let \( f: X \to \mathbb{R} \) be a function.

We say that \( f \) converges to \( L \) as \( x \to +\infty \) in \( X \), and write \( \lim_{x \to +\infty} f(x) = L \) iff

for any real \( \varepsilon > 0 \) there exists an \( M \in X \) such that \( |f(x) - L| \le \varepsilon \) for all \( x \in X \text{ such that } x > M \).

A similar formulation can be made for \( x \to -\infty \).



Source

p250-251