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 $f : X \to R$ , where $E \subseteq X \subseteq R$ , 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 $X \subseteq R$ .

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 $X \subseteq R$ with $+ \infty$ being an adherent point, and let $f : X \to R$ 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 $x \to - \infty$ .

19.03.2020