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 [...].
We say that \( -\infty \) is adherent to \( X \) iff [...].
In other words, \( +\infty \) is adherent to \( X \) iff \( X \) has no upper bound, or equivalently, [ \( ? = +\infty \)]. Similarly, \( -\infty \) is adherent to \( X \) iff \( X \) has no lower bound, or equivalently, [\( ? = -\infty \)].
So a set is [...] 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
[...]
A similar formulation can be made for \( x \to -\infty \).