Equivalent infinitesimals and infinitely large functions

Summary

This lesson closes the unit on limits with a computational technique that brings together almost everything that came before it: the comparison of functions that tend to zero or to infinity. Little $o$ , big $O$ , the same order and equivalence are defined, and it is proved that, in a product or a quotient, each factor can be replaced by an equivalent one without changing the limit. The trigonometric limits, those of the exponential, the logarithm and powers, and those of polynomials, proved in the preceding lessons, are gathered into a table of equivalents, which makes it possible to compute at a glance limits that used to require several transformations. The lesson also shows the most common error with this technique, substituting inside a sum, when a sum can indeed be substituted, and how powers compare near zero and far from the origin.

Learning objectives

Define little $o$ , big $O$ , the same order and the equivalence of functions at a point, on one side or at infinity, and prove their properties.
Prove the theorem on substitution of equivalents in products and quotients, and recognize why it does not hold for sums.
Prove the table of equivalents at zero and use it, together with composition, to compute limits and to determine the order and the principal part of an infinitesimal.
Compare infinitely large powers and infinitesimal powers, and distinguish equivalence from the existence of an asymptote.

Comparing functions

The lessons on the squeeze theorem and on infinite limits showed that the forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$ have no fixed result: the limit of a quotient of two functions that tend to zero depends on how fast they do so. The idea of this lesson is to measure that speed by comparing each function with a simpler one. As in the lesson on infinite limits, $P$ denotes the target of the variable ( $a$ , $a^+$ , $a^-$ , $\infty$ or $-\infty$ ), and “near $P$ ” means in a punctured interval, on one side or on a ray. If a function is defined only on a set $A$ of which $a$ is an accumulation point, “near $a$ ” means at the points of $A$ in a punctured interval centered at $a$ , and the limits are those of the lesson on the limit of functions of a real variable, taken in $A$ ; the following definitions apply in the same way in that case.

DefinitionComparison of functions

Let $f$ and $g$ be functions defined near $P$ , with $g(x) \neq 0$ near $P$ .

$f$ is negligible compared with $g$ at $P$ , and one writes $f = \operatorname{o}(g)$ , which is read “ $f$ is little-o of $g$ ”, if $\lim_{x \to P} \frac{f(x)}{g(x)} = 0$ .
$f$ is bounded by $g$ at $P$ , and one writes $f = \operatorname{O}(g)$ , which is read “ $f$ is big-O of $g$ ”, if there exists $C > 0$ such that $\abs{f(x)} \leq C\abs{g(x)}$ near $P$ ; that is, if the quotient $\frac{f(x)}{g(x)}$ is bounded near $P$ .
$f$ and $g$ are of the same order at $P$ if $\frac{f(x)}{g(x)}$ tends to a nonzero number.
$f$ and $g$ are equivalent at $P$ , and one writes $f \sim g$ , if $\lim_{x \to P} \frac{f(x)}{g(x)} = 1$ .

A function that tends to $0$ at $P$ is called an infinitesimal at $P$ , and one that tends to $\infty$ or to $-\infty$ , an infinitely large function. An infinitesimal at $0$ has order $k > 0$ if it is of the same order as $x^k$ (at $0^+$ , if $x^k$ is defined only for $x > 0$ ), and then, if $\frac{f(x)}{x^k}$ tends to $c$ , the function $c\,x^k$ is its principal part: $f \sim c\,x^k$ .

The word “infinitesimal” does not denote a very small number, but a function that tends to zero; the notation $\operatorname{o}(g)$ does not denote a specific function, but any of those that are negligible compared with $g$ . For example, $x^2 = \operatorname{o}(x)$ at $0$ , because $\frac{x^2}{x} = x$ tends to $0$ ; on the other hand, $x = \operatorname{o}(x^2)$ at $\infty$ , because $\frac{x}{x^2} = \frac{1}{x}$ tends to $0$ . One and the same pair of functions compares in opposite ways depending on the target. Being of the same order and being equivalent are, at bottom, a single comparison: if $f$ and $g$ are of the same order with a quotient that tends to $c$ , then $f \sim c\,g$ .

PropositionProperties of the comparison

At one and the same target $P$ :

equivalence is reflexive, symmetric and transitive: $f \sim f$ ; if $f \sim g$ , then $g \sim f$ ; and if $f \sim g$ and $g \sim h$ , then $f \sim h$ ;
$f \sim g$ if and only if $f - g = \operatorname{o}(g)$ ; in particular, if $h = \operatorname{o}(g)$ , then $g + h \sim g$ ;
if $f = \operatorname{o}(h)$ and $g = \operatorname{o}(h)$ , then $f + g = \operatorname{o}(h)$ and $c\,f = \operatorname{o}(h)$ for every real $c$ .

Each statement is a rule of the algebra of limits applied to a quotient.

Proof

$\frac{g}{f} = \frac{1}{\frac{f}{g}} \to 1, \qquad \frac{f}{h} = \frac{f}{g} \cdot \frac{g}{h} \to 1 \cdot 1 = \resaltar{1}$
The quotient of $f$ by itself equals $1$ . If $\frac{f}{g}$ tends to $1$ , its reciprocal tends to $\frac{1}{1} = 1$ , by the rule for the reciprocal. If $\frac{f}{g}$ and $\frac{g}{h}$ tend to $1$ , so does their product, by the product rule.
$\frac{f - g}{g} = \frac{f}{g} - 1 \to 0 \Leftrightarrow \frac{f}{g} \to \resaltar{1}$
The quotient of $f - g$ by $g$ is $\frac{f}{g} - 1$ , which tends to $0$ if and only if $\frac{f}{g}$ tends to $1$ . With $f = g + h$ , the difference is $h$ .
$\frac{f + g}{h} = \frac{f}{h} + \frac{g}{h} \to 0 + 0 = \resaltar{0}$
If $\frac{f}{h}$ and $\frac{g}{h}$ tend to $0$ , so do their sum and the product of each of them by a constant, by the rule for linear combinations.

Big $O$ does not require the quotient to have a limit, only that it not grow out of control. For example, $\sin x = \operatorname{O}(1)$ at $\infty$ , even though $\sin x$ has no limit there, and $x \sin \frac{1}{x} = \operatorname{O}(x)$ at $0$ , because $\abs{x \sin \frac{1}{x}} \leq \abs{x}$ . To compute limits, little $o$ is enough; big $O$ will serve to measure approximation errors, such as that of the Taylor polynomial, which will be studied in Calculus II. Its rules are the following.

PropositionRules for big O

At one and the same target $P$ :

if $\frac{f}{g}$ has a finite limit at $P$ , then $f = \operatorname{O}(g)$ ; in particular, if $f = \operatorname{o}(g)$ or if $f \sim g$ ;
if $f = \operatorname{O}(h)$ and $g = \operatorname{O}(h)$ , then $f + g = \operatorname{O}(h)$ and $c\,f = \operatorname{O}(h)$ for every real $c$ ;
if $f_1 = \operatorname{O}(g_1)$ and $f_2 = \operatorname{O}(g_2)$ , then $f_1 f_2 = \operatorname{O}(g_1 g_2)$ ;
if $f_1 = \operatorname{O}(g_1)$ and $f_2 = \operatorname{o}(g_2)$ , then $f_1 f_2 = \operatorname{o}(g_1 g_2)$ .

Each rule is an inequality between absolute values; the last one combines a bound with a quotient that tends to zero.

Proof

$\abs{\frac{f(x)}{g(x)}} \leq \abs{\frac{f(x)}{g(x)} - L} + \abs{L} < \resaltar{\abs{L} + 1}$
If $\frac{f}{g}$ tends to $L$ , the challenge $\varepsilon = 1$ has a response: near $P$ , the distance from the quotient to $L$ is less than $1$ , and by the triangle inequality its absolute value does not exceed $\abs{L} + 1$ .
$\abs{f(x) + g(x)} \leq C_1\abs{h(x)} + C_2\abs{h(x)} = \resaltar{(C_1 + C_2)\abs{h(x)}}$
The bounds $C_1$ of $f$ and $C_2$ of $g$ hold simultaneously near $P$ , in the intersection of the two neighborhoods. The triangle inequality bounds the sum; the multiple has the bound $\abs{c}\,C_1$ .
$\abs{f_1(x) f_2(x)} \leq C_1\abs{g_1(x)} \cdot C_2\abs{g_2(x)} = \resaltar{C_1 C_2 \abs{g_1(x) g_2(x)}}$
The bounds of the two factors are multiplied.
$0 \leq \abs{\frac{f_1(x) f_2(x)}{g_1(x) g_2(x)}} \leq C_1 \abs{\frac{f_2(x)}{g_2(x)}} \to \resaltar{0}$
The quotient of the product by $g_1 g_2$ is the product of a quotient bounded by $C_1$ and another that tends to $0$ ; by the squeeze theorem, it tends to $0$ .