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 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, $i n f (X) = - \infty$ .

So a set is bounded 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

for any real $ε > 0$ there exists an $M \in X$ such that $| f (x) - L | \leq ε$ for all $x \in X such that x > M$ .

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

Source

p250-251

19.03.2020