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 $$-\infty$$.

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 $$x \to -\infty$$.