One-sided limits · Topos Uranos

Summary

This lesson studies the behavior of a function on each side of a point. The right-hand limit and the left-hand limit are defined as ordinary limits of the restriction of the function to one of the sides, so that all the theorems of the preceding lessons hold for them without a new proof. Only two results are proved with epsilon and delta: that a function has a limit if and only if its two one-sided limits exist and coincide, and that a monotonic bounded function has a one-sided limit at an endpoint of an interval, equal to the supremum or the infimum of its values. The latter is the first theorem of the course that guarantees that a limit exists without knowing its value in advance. The problems study piecewise-defined functions, the integer part, and trigonometric quotients that tend to different numbers on each side.

Learning objectives

Define one-sided limits with epsilon and delta as limits of a restriction, and apply to them the theorems of the preceding lessons.
Prove that a function has a limit at a point if and only if its two one-sided limits exist and are equal, and use this to prove that a limit does not exist.
Prove that a monotonic bounded function has one-sided limits at the endpoints of an interval, equal to the supremum or the infimum of its values.
Compute one-sided limits of piecewise-defined functions, of the integer part and of trigonometric quotients, and determine constants so that a limit exists.

Jumps and edges

In the lesson on the limit of functions of a real variable it was proved that the sign function, which equals $1$ on the positive numbers and $-1$ on the negative ones (the value assigned to it at $0$ does not affect the limit), has no limit at $0$ . And yet its behavior near $0$ is very simple: to the right of $0$ it is constantly $1$ , and to the left, constantly $-1$ . The definition of limit does not distinguish the sides, and that is why it does not record this information. The same happens with the integer part, defined below, at each integer.

In the proof of the density of the rational numbers, in the lesson on the real line, it was shown that for every real $t$ there exists an integer $m$ with $m - 1 \leq t < m$ . This integer is unique, because two distinct integers are at least $1$ apart and do not both fit in an interval of length $1$ open at one end. The integer $m - 1$ is called the integer part of $t$ and is written $\lfloor t \rfloor$ : it is the only integer that satisfies the following condition.

\lfloor t \rfloor \in \mathbb{Z}, \qquad \lfloor t \rfloor \leq t < \lfloor t \rfloor + 1

For example, $\lfloor 2.7 \rfloor = 2$ , $\lfloor 3 \rfloor = 3$ and $\lfloor -0.5 \rfloor = -1$ . On the interval $\ico{n}{n + 1}$ , with $n$ an integer, the integer part is constantly $n$ ; on reaching $n + 1$ it jumps by one. Near an integer $n$ , the function equals $n - 1$ on the left and $n$ on the right.

Throughout the lesson, $f$ is a real function defined on a set $A$ , and we write $A \cap \ioo{a}{\infty}$ for the points of $A$ lying to the right of $a$ and $A \cap \ioo{-\infty}{a}$ for those lying to its left.