Algebra of limits · Topos Uranos

Summary

This lesson derives from the definition of limit the rules that make it possible to compute limits without searching each time for a $\delta$ . The proofs are ordered so that each one takes advantage of the earlier ones: three lemmas, proved by epsilon-delta arguments, reduce every rule to an inequality, and with them the limits of the sum, the product, the reciprocal and the quotient are obtained. From these follow the limits of polynomials and rational functions, which justify direct substitution; then those of roots and powers with rational exponents, and the theorem on the limit of a composition, which justifies the change of variable. The lesson ends with what the rules do not say: when a quotient has no finite limit and why the rules do not hold in the reverse direction. The problems resolve $\frac{0}{0}$ forms by factoring, rationalizing and changing the variable, and determine constants so that a limit exists.

Learning objectives

Prove the bound criterion by an epsilon-delta argument and deduce from it the limits of the sum, the product, the reciprocal and the quotient.
Prove the limits of polynomials, rational functions, roots and powers with rational exponents, and apply direct substitution with its conditions.
Prove the theorem on the limit of a composition, recognize why its hypotheses are needed and apply it as a change of variable.
Compute limits of the form $\frac{0}{0}$ by factoring, rationalizing or changing the variable, and prove when a quotient has no finite limit.

The plan of the proofs

In the lesson on the limit of functions of a real variable it was proved from the definition that $\lim_{x \to 3} x^2 = 9$ , and the work consisted in bounding $\abs{x^2 - 9}$ by a multiple of $\abs{x - 3}$ . All the rules of this lesson have that same form: to prove that a function $F$ tends to $K$ , one bounds the distance $\abs{F(x) - K}$ by a multiple of distances that are already known to become small. If this is so, it is worth proving once and for all, by an epsilon-delta argument, that such a bound suffices; afterwards, each rule reduces to finding the appropriate inequality, and no epsilon-delta argument is repeated.

Throughout the lesson, $f$ and $g$ are real functions defined on the same set $A$ , the point $a$ is an accumulation point of $A$ and, when we say that $f(x)$ satisfies something “near $a$ ”, we mean that there exists an $r > 0$ such that every $x$ in $A$ with $0 < \abs{x - a} < r$ satisfies it. From the preceding lesson we use the definition, the uniqueness of the limit, the limits of constant functions and of the identity function, the lemma according to which the limit depends only on the values near the point, and local boundedness together with preservation of sign.