Limits and continuity: synthesis and challenges

Summary

This lesson closes the unit on limits and continuity. It gathers into a single deductive chain what the ten lessons of the unit proved: the epsilon-delta definition of the limit and its local properties, the rules of the algebra of limits reduced to three lemmas, the squeeze theorem and the trigonometric limits; one-sided limits, limits at infinity and infinite limits, which inherit those rules by restriction, by inversion and by the reciprocal, together with the asymptotes; and continuity, the continuity of the elementary functions, the standard limits and the comparison of functions by means of equivalent infinitesimals. It explains how the unit rests on Algebra and Geometry I and II, whose admitted facts about asymptotes and continuity it turns into theorems, and how it prepares the study of derivatives and their applications. It ends with an exit assessment, parallel to the entry assessment, and with ten challenges, harder than the problems of the lessons, each of which combines results from several of them.

Learning objectives

Set out in logical order the definitions and the main theorems of the unit, indicating on which earlier results each one depends and which epsilon-delta arguments are reused.
Distinguish what the unit admits from what it proves, and recognize the hypotheses of each result, such as that of the limit of a composition or that of the substitution of equivalents.
Relate the results of the unit to Algebra and Geometry I and II and to the study of derivatives and their applications.
Solve non-routine problems that combine the algebra of limits, one-sided limits, limits at infinity and infinite limits, asymptotes, continuity, the standard limits and equivalent infinitesimals.

What the unit achieved

The unit started from a question and a demand. The question: what it means for the values of a function to approach a number as the variable approaches a point. The demand: that the answer should make it possible to prove, and not only to compute, and that each epsilon-delta argument should be carried out only once and then reused. The answer was a single definition, stated in the lesson on the limit of functions of a real variable: if $a$ is an accumulation point of the domain $A$ of $f$ , the statement that the limit of $f$ at $a$ is $L$ means the following.

(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x \in A)\left(0 < \abs{x - a} < \delta \rightarrow \abs{f(x) - L} < \varepsilon\right)

Everything else was obtained from it, by reduction: the computation rules, to an inequality; one-sided limits, to limits of restrictions; limits at infinity, to one-sided limits at zero; infinite limits, to zero limits of the reciprocals; and continuity, to the limit being equal to the value.

The definition and algebra of the limit

The first lesson read the definition as a game of challenge and response, wrote it as a quantified statement and obtained its negation quantifier by quantifier. With the requirement that $a$ be an accumulation point of the domain it proved the uniqueness of the limit; and with challenges chosen in advance, that the limit depends only on the values near the point, that it forces the function to remain bounded near it and that, if it is not zero, it imposes its sign on the function. The lesson on the algebra of limits arranged the proofs so as to reuse them: three lemmas, proved with epsilon and delta, show that every function $F$ that satisfies near $a$ a bound of the form $\abs{F(x) - K} \leq C\left(\abs{f(x) - L} + \abs{g(x) - M}\right)$ tends to $K$ , and each rule was reduced to finding such a bound: the one for linear combinations follows from the triangle inequality, the one for products from local boundedness and the one for reciprocals from the preservation of sign, with a lemma ensuring that restricting the domain does not lose the limit. From there followed direct substitution in polynomials and rational functions, the limits of roots and of powers with rational exponents, and the theorem on composition, which justifies the change of variable under a hypothesis that a counterexample shows to be necessary: that the inner function does not take the value of its limit near the point. The lesson on the squeeze theorem proved that limits preserve non-strict inequalities, that a function trapped between two functions that tend to the same number also tends to it, and that the product of a bounded function and one that tends to zero tends to zero; with the bound $\abs{\sin x - \sin a} \leq \abs{x - a}$ and the inequality between the sine, the arc and the tangent, it obtained the fundamental limit $\frac{\sin x}{x} \to 1$ and its consequences.

The sides, infinity and the asymptotes

The lesson on one-sided limits defined them as limits of the restriction of the function to one side of the point. That decision saved all the remaining work: the theorems already proved hold for them without a new proof, and the limit exists if and only if both one-sided limits exist and are equal. The same lesson proved, for the first time without computing the value, that a limit exists: a function that is increasing and bounded above on an interval tends, at its right endpoint, to the supremum of its values, and completeness guarantees it. The lesson on limits as the variable tends to infinity reduced each of them, with the inversion lemma, to a one-sided limit at zero, and thus obtained the behavior of rational functions far from the origin. The lesson on infinite limits completed the scheme with the case in which the values grow without bound: the reciprocal lemma translates them into zero limits, so that the rules summarized as “ $\infty + c = \infty$ ” or “ $c \cdot \infty = \infty$ ” are theorems with hypotheses, and the four indeterminate forms are warnings, not results. Finally, the lesson on asymptotes defined them by means of limits, found the slope and the intercept of a non-vertical asymptote as two limits, whence there is at most one at each end, proved the vertical asymptotes of rational functions and organized the sketch of a graph without derivatives; it also showed that a curve can cross its asymptote.

Continuity and the comparison of functions

The lesson on limit and continuity defined continuity at a point of the domain, proved that at an accumulation point it is equivalent to the limit being the value, and classified discontinuities into removable, jump and essential ones. The rules of the algebra of limits gave the continuity of the operations, and the composition of continuous functions turned out to be continuous without the hypothesis that the theorem on the limit of a composition required; it follows that the limit passes inside a continuous function. The lesson on the continuity of the elementary functions proved with a single argument, that of monotonic functions whose image is an interval, the continuity of the exponential, of the logarithm, of powers with real exponents and of the inverse trigonometric functions; with it and the inequality $e^x \geq 1 + x$ it obtained the standard limits, the number $e$ as a limit and the form $1^\infty$ , and compared the exponential, powers and the logarithm at infinity. The lesson on equivalent infinitesimals and infinitely large functions read all those limits as comparisons between functions: it defined little $\operatorname{o}$ , big $\operatorname{O}$ and equivalence, proved that in a product or a quotient each factor can be replaced by an equivalent one, and gathered the limits obtained before into a table of equivalents at zero.

What is admitted, what is proved and what remains

The balance of the unit can be summarized as follows:

A single axiom is admitted: completeness, together with the field and order axioms, according to which every nonempty set of reals that is bounded above has a supremum. In addition, results proved in Algebra and Geometry are used: the inequality between the sine, the arc and the tangent, the sum-to-product formulas, the inequality $e^x \geq 1 + x$ , the monotonicity of the exponential and the factor theorem.
Facts that Algebra and Geometry had admitted are proved: the vertical asymptotes and the asymptotes at infinity of rational functions, and the continuity of the rational, trigonometric and inverse trigonometric functions.
Five tools are reused instead of repeating arguments: the three lemmas of the bound criterion, the preservation of sign, the restriction of the domain (in particular, to one side of the point), inversion and the reciprocal.
What requires more than limits is left for later: the intermediate value theorem and derivatives, in the following unit; the Weierstrass theorem and L’Hôpital’s rule, in the third; and the Taylor polynomial, in Calculus II.

The following figure gathers the three segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that come from Algebra and Geometry, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

AAxioms of the reals. It is admitted, from Algebra and Geometry I, that the reals form a complete ordered field: addition and multiplication satisfy the field axioms, the order satisfies trichotomy and is compatible with the operations, and every nonempty set of reals that is bounded above has a supremum. Admitted without proof in this unit.
BInheritance from Algebra and Geometry. Results proved in Algebra and Geometry I and II: the factorization of $u^n - v^n$ and the power with a rational exponent, $\abs{\sin x} \leq \abs{x}$ and $\sin x < x < \tan x$ if $0 < x < \frac{\pi}{2}$ , the sum-to-product and power-reducing formulas, the inequality $e^x \geq 1 + x$ , the monotonicity and the range of the exponential, the inverse trigonometric functions, the factor theorem and sign charts. Admitted without proof in this unit.
CBound criterion. If near $a$ we have $\abs{F(x) - K} \leq C\left(\abs{f(x) - L} + \abs{g(x) - M}\right)$ , with $C > 0$ , $f \to L$ and $g \to M$ , then $F \to K$ ; each rule reduces to finding such a bound. Established in: 4. Three lemmas.
DLocal boundedness and sign. If $f$ tends to $L$ at $a$ , then near $a$ we have $\abs{f(x)} < \abs{L} + 1$ ; and if $L \neq 0$ , near $a$ the function has the sign of $L$ and $\abs{f(x)} > \frac{\abs{L}}{2}$ . Established in: 3. Local properties.
EChange of variable. If $g(x) \to b$ , $f(t) \to L$ as $t \to b$ , and $g(x) \neq b$ near $a$ , then $f(g(x)) \to L$ . Established in: 8. Limit of a composition.
FRestriction. If $f$ tends to $L$ at $a$ and $a$ is an accumulation point of a subset $B$ of the domain, the restriction of $f$ to $B$ also tends to $L$ ; a one-sided limit is the limit of a restriction, and that is why all theorems on limits hold for one-sided limits. Established in: 5. Limit of a restriction.
GInversion. The limit of $f$ at $\infty$ is the limit of $f\left(\frac{1}{t}\right)$ as $t$ tends to $0$ from the right, and the limit at $-\infty$ , from the left; that is why all theorems on limits hold at infinity. Established in: 15. Inversion lemma.
HReciprocal. A function tends to $\infty$ if and only if it is positive near the target and its reciprocal tends to $0$ ; that is why the rules for finite limits decide the infinite ones. Established in: 17. Reciprocal lemma.

Segment 1 The definition and algebra of the limit
- Define the limit at an accumulation point with epsilon and delta, write it and negate it as a quantified statement, and prove its uniqueness and its local properties.
- Reduce each rule of the algebra of limits to a bound by means of three lemmas, and obtain the limits of rational functions, roots and powers.
- Change the variable with the theorem on composition, recognizing its hypothesis.
- Prove the squeeze theorem and, with it, the fundamental trigonometric limit and its consequences.
1. Definition of limit. If $a$ is an accumulation point of the domain $A$ of $f$ , the limit of $f$ at $a$ is $L$ when $(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x \in A)\left(0 < \abs{x - a} < \delta \rightarrow \abs{f(x) - L} < \varepsilon\right)$ ; and it is not when $(\exists \varepsilon > 0)(\forall \delta > 0)(\exists x \in A)\left(0 < \abs{x - a} < \delta \wedge \abs{f(x) - L} \geq \varepsilon\right)$ .
  Proved in: Limit of functions of a real variable.
2. Uniqueness of the limit. If $a$ is an accumulation point of the domain of $f$ , $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} f(x) = L'$ , then $L = L'$ .
  Uses: 1. Definition of limit. Rests on: A (Axioms of the reals). Proved in: Limit of functions of a real variable.
3. Local properties. The limit at $a$ depends only on the values of the function on a punctured interval with center $a$ ; if $\lim_{x \to a} f(x) = L$ , then near $a$ we have $\abs{f(x)} < \abs{L} + 1$ , and if moreover $L \neq 0$ , $f(x)$ has the sign of $L$ and $\abs{f(x)} > \frac{\abs{L}}{2}$ .
  Uses: 1. Definition of limit. Rests on: A (Axioms of the reals). Proved in: Limit of functions of a real variable.
4. Three lemmas. The limit of $f$ at $a$ is $L$ if and only if the limit of $\abs{f(x) - L}$ is $0$ ; if $\abs{f(x) - L} \leq C\abs{h(x)}$ near $a$ , with $C > 0$ and $\lim_{x \to a} h(x) = 0$ , then $\lim_{x \to a} f(x) = L$ ; and the sum of two distances that tend to zero tends to zero.
  Uses: 1. Definition of limit. Proved in: Algebra of limits.
5. Limit of a restriction. If $B$ is a subset of the domain $A$ of $f$ , $a$ is an accumulation point of $B$ and $\lim_{x \to a} f(x) = L$ , then the restriction of $f$ to $B$ also tends to $L$ at $a$ : the response that works in $A$ works in $B$ .
  Uses: 1. Definition of limit. Proved in: Algebra of limits.
6. Algebra of limits. If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ , then $\alpha f + \beta g$ tends to $\alpha L + \beta M$ and $fg$ to $LM$ ; and if $M \neq 0$ , $\frac{1}{g}$ tends to $\frac{1}{M}$ and $\frac{f}{g}$ to $\frac{L}{M}$ .
  Uses: 3. Local properties, 4. Three lemmas. Rests on: C (Bound criterion), D (Local boundedness and sign), F (Restriction). Proved in: Algebra of limits.
7. Polynomials, roots and powers. A polynomial, and a rational function whose denominator does not vanish at $a$ , have their value as their limit at $a$ ; $\sqrt[n]{f(x)}$ tends to $\sqrt[n]{L}$ , with $f \geq 0$ if $n$ is even; and if $L > 0$ , $\left(f(x)\right)^{\frac{p}{q}}$ tends to $L^{\frac{p}{q}}$ .
  Uses: 1. Definition of limit, 6. Algebra of limits. Rests on: B (Inheritance from Algebra and Geometry), C (Bound criterion), D (Local boundedness and sign). Proved in: Algebra of limits.
8. Limit of a composition. If $\lim_{x \to a} g(x) = b$ , $\lim_{t \to b} f(t) = L$ and $g(x) \neq b$ near $a$ , then $\lim_{x \to a} f(g(x)) = L$ ; without the hypothesis $g(x) \neq b$ , the conclusion can fail.
  Uses: 1. Definition of limit. Proved in: Algebra of limits.
9. Preservation of inequalities. If $f(x) \leq g(x)$ near $a$ , $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ , then $L \leq M$ ; the strict inequality is not preserved.
  Uses: 6. Algebra of limits. Rests on: D (Local boundedness and sign). Proved in: The squeeze theorem and trigonometric limits.
10. Squeeze theorem. If $h(x) \leq f(x) \leq g(x)$ near $a$ and $h$ and $g$ tend to $L$ , then $f$ also tends to $L$ ; and the product of a function bounded near $a$ and one that tends to $0$ tends to $0$ .
  Uses: 4. Three lemmas. Rests on: C (Bound criterion). Proved in: The squeeze theorem and trigonometric limits.
11. Sine and cosine. For all reals $x$ and $a$ , $\abs{\sin x - \sin a} \leq \abs{x - a}$ and $\abs{\cos x - \cos a} \leq \abs{x - a}$ ; therefore, $\lim_{x \to a} \sin x = \sin a$ and $\lim_{x \to a} \cos x = \cos a$ .
  Uses: 4. Three lemmas. Rests on: B (Inheritance from Algebra and Geometry), C (Bound criterion). Proved in: The squeeze theorem and trigonometric limits.
12. Fundamental limit. $\lim_{x \to 0} \frac{\sin x}{x} = 1$ ; and as $x$ tends to $0$ , $\frac{\tan x}{x}$ tends to $1$ , $\frac{\sin kx}{x}$ to $k$ and $\frac{1 - \cos x}{x^2}$ to $\frac{1}{2}$ .
  Uses: 6. Algebra of limits, 10. Squeeze theorem, 11. Sine and cosine. Rests on: B (Inheritance from Algebra and Geometry), E (Change of variable). Proved in: The squeeze theorem and trigonometric limits.
Segment 2 The sides, infinity and the asymptotes
- Define one-sided limits as limits of restrictions, so that they inherit all the theorems, and characterize the limit by its two sides.
- Obtain from completeness the limit of a monotonic and bounded function as a supremum.
- Reduce limits at infinity to one-sided limits at zero, and infinite limits to zero limits, with their rules and their indeterminate forms.
- Compute the asymptotes of a function with two limits, prove the vertical asymptotes of rational functions and sketch graphs without derivatives.
1. One-sided limits. The right-hand limit of $f$ at $a$ is the limit at $a$ of its restriction to the points of the domain greater than $a$ , and the left-hand limit, to the smaller ones; if $a$ is an accumulation point from both sides, $f$ has limit $L$ at $a$ if and only if both one-sided limits exist and equal $L$ .
  Uses: 1. Definition of limit, 5. Limit of a restriction, 2. Uniqueness of the limit. Proved in: One-sided limits.
2. Monotonic functions. If $f$ is increasing and bounded above on $\ioo{c}{b}$ , then $\lim_{x \to b^-} f(x) = \sup\left\{ f(x) : c < x < b \right\}$ ; and if it is increasing and bounded below, its right-hand limit at $c$ is the infimum of its values.
  Uses: 6. Algebra of limits, 13. One-sided limits. Rests on: A (Axioms of the reals), E (Change of variable), F (Restriction). Proved in: One-sided limits.
3. Inversion lemma. The limit of $f$ at $\infty$ is $L$ if for every $\varepsilon > 0$ there exists $M$ such that $\abs{f(x) - L} < \varepsilon$ whenever $x > M$ ; and $\lim_{x \to \infty} f(x) = L \Leftrightarrow \lim_{t \to 0^+} f\left(\frac{1}{t}\right) = L$ holds, and likewise at $-\infty$ with the left-hand limit.
  Uses: 13. One-sided limits. Proved in: Limits as the variable tends to infinity.
4. Rational functions at infinity. For every rational $r > 0$ , $\frac{1}{x^r}$ tends to $0$ at $\infty$ ; and the quotient of two polynomials of degrees $n$ and $m$ tends, at $\pm\infty$ , to $0$ if $n < m$ , to the quotient of the leading coefficients if $n = m$ , and has no finite limit if $n > m$ .
  Uses: 2. Uniqueness of the limit, 6. Algebra of limits, 7. Polynomials, roots and powers. Rests on: D (Local boundedness and sign), F (Restriction), G (Inversion). Proved in: Limits as the variable tends to infinity.
5. Reciprocal lemma. $f$ tends to $\infty$ at a target $P$ , which is a point, one side of a point or $\pm\infty$ , if for every real $N$ we have $f(x) > N$ near $P$ ; this happens if and only if $f$ is positive near $P$ and $\frac{1}{f}$ tends to $0$ ; and $f$ tends to $-\infty$ if and only if $-f$ tends to $\infty$ .
  Uses: 1. Definition of limit. Proved in: Infinite limits.
6. Rules with infinite limits. If $f \leq g$ near $P$ and $f$ tends to $\infty$ , so does $g$ ; if $f$ tends to $\infty$ , so does $f + g$ if $g$ is bounded below, and so does $fg$ if $g$ tends to a positive number or to $\infty$ ; and a quotient whose numerator tends to a positive number and whose denominator tends to $0$ through positive values tends to $\infty$ .
  Uses: 6. Algebra of limits. Rests on: D (Local boundedness and sign), F (Restriction), G (Inversion), H (Reciprocal). Proved in: Infinite limits.
7. Unbounded monotonic functions. If $f$ is increasing on $\ioo{c}{b}$ , with $b$ real or $b = \infty$ , and is not bounded above, then $f$ tends to $\infty$ from the left at $b$ , or at $\infty$ if $b = \infty$ .
  Uses: 17. Reciprocal lemma. Proved in: Infinite limits.
8. Composition with an infinite limit. If $g$ tends to $\infty$ at $P$ and $f(u)$ tends to $L$ as $u$ tends to $\infty$ , then $f(g(x))$ tends to $L$ at $P$ , without any hypothesis on the values of $g$ ; and likewise if $f(u)$ tends to $\infty$ or to $-\infty$ .
  Uses: 15. Inversion lemma, 17. Reciprocal lemma. Proved in: Infinite limits.
9. Computing the asymptote. The line $y = mx + b$ is an asymptote of the graph of $f$ at $\infty$ , that is, $f(x) - (mx + b)$ tends to $0$ , if and only if $m = \lim_{x \to \infty} \frac{f(x)}{x}$ and $b = \lim_{x \to \infty} \left(f(x) - mx\right)$ ; therefore, at each end there is at most one non-vertical asymptote.
  Uses: 2. Uniqueness of the limit, 6. Algebra of limits, 16. Rational functions at infinity. Rests on: G (Inversion). Proved in: Asymptotes and curve sketching techniques.
10. Vertical asymptotes. If $r$ is a root of the polynomial $B$ and $A(r) \neq 0$ , the line $x = r$ is a vertical asymptote of the graph of $\frac{A}{B}$ , and on each side of $r$ the function tends to $\infty$ or to $-\infty$ according to the sign that the sign chart assigns to it on the adjacent interval.
  Uses: 7. Polynomials, roots and powers, 18. Rules with infinite limits. Rests on: B (Inheritance from Algebra and Geometry), D (Local boundedness and sign), F (Restriction). Proved in: Asymptotes and curve sketching techniques.
Segment 3 Continuity and the comparison of functions
- Define continuity, relate it to the limit and classify discontinuities into removable, jump and essential ones.
- Prove that the operations and composition preserve continuity, and that the limit passes inside a continuous function.
- Prove the continuity of the elementary functions with the argument of monotonic functions and, with it, the standard limits and the form $1^\infty$ .
- Compare functions with little $\operatorname{o}$ and equivalence, and substitute equivalents in products and quotients, but not in sums.
1. Continuity and limit. $f$ is continuous at a point $a$ of its domain $A$ if $(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x \in A)\left(\abs{x - a} < \delta \rightarrow \abs{f(x) - f(a)} < \varepsilon\right)$ ; if $a$ is an accumulation point of $A$ , this is equivalent to $\lim_{x \to a} f(x) = f(a)$ , and if it is an isolated point, it always holds.
  Uses: 1. Definition of limit. Proved in: Limit and continuity.
2. Operations with continuous functions. Linear combinations, products, quotients with a nonzero denominator and roots of functions continuous at $a$ are continuous at $a$ ; therefore, polynomials, rational functions, roots, the absolute value, the sine, the cosine and the tangent are continuous on their domain.
  Uses: 6. Algebra of limits, 7. Polynomials, roots and powers, 11. Sine and cosine, 23. Continuity and limit. Rests on: C (Bound criterion). Proved in: Limit and continuity.
3. Composition of continuous functions. If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$ , then $f \circ g$ is continuous at $a$ ; and if $g(x)$ tends to $b$ and $f$ is continuous at $b$ , then $\lim_{x \to a} f(g(x)) = f(b)$ , also from one side or at $\pm\infty$ .
  Uses: 3. Local properties, 23. Continuity and limit. Rests on: F (Restriction), G (Inversion). Proved in: Limit and continuity.
4. Continuous elementary functions. A function defined on an interval, monotonic on it and whose image is an interval is continuous, and so is the inverse of a strictly monotonic function with those properties; hence the exponential, the logarithm, powers with real exponents, the inverse trigonometric functions and the hyperbolic functions are continuous on their domain.
  Uses: 13. One-sided limits, 14. Monotonic functions, 23. Continuity and limit, 24. Operations with continuous functions, 25. Composition of continuous functions. Rests on: B (Inheritance from Algebra and Geometry). Proved in: Continuity of the elementary functions and standard limits.
5. Standard limits. As $x$ tends to $0$ , $\frac{e^x - 1}{x}$ and $\frac{\ln(1 + x)}{x}$ tend to $1$ , $\frac{a^x - 1}{x}$ to $\ln a$ and $\frac{\left(1 + x\right)^r - 1}{x}$ to $r$ ; moreover, $\left(1 + x\right)^{\frac{1}{x}}$ tends to $e$ at $0$ , and $\left(1 + \frac{1}{x}\right)^x$ does so at $\pm\infty$ .
  Uses: 6. Algebra of limits, 10. Squeeze theorem, 13. One-sided limits, 26. Continuous elementary functions, 25. Composition of continuous functions. Rests on: B (Inheritance from Algebra and Geometry), E (Change of variable), G (Inversion). Proved in: Continuity of the elementary functions and standard limits.
6. The form one to the power infinity. If $f > 0$ near $P$ , $f(x)$ tends to $1$ and $g(x)\left(f(x) - 1\right)$ tends to a real number $\lambda$ , then $\left(f(x)\right)^{g(x)}$ tends to $e^{\lambda}$ .
  Uses: 6. Algebra of limits, 26. Continuous elementary functions, 25. Composition of continuous functions, 27. Standard limits. Proved in: Continuity of the elementary functions and standard limits.
7. Exponentials, powers and logarithms. If $r > 0$ and $a > 1$ , then $\frac{x^r}{a^x}$ and $\frac{\ln x}{x^r}$ tend to $0$ at $\infty$ , and $x^r \ln x$ tends to $0$ as $x$ tends to $0$ from the right.
  Uses: 10. Squeeze theorem, 16. Rational functions at infinity, 19. Unbounded monotonic functions, 20. Composition with an infinite limit. Rests on: B (Inheritance from Algebra and Geometry), G (Inversion). Proved in: Continuity of the elementary functions and standard limits.
8. Comparison of functions. $f = \operatorname{o}(g)$ at $P$ if $\frac{f}{g}$ tends to $0$ , and $f \sim g$ if it tends to $1$ ; equivalence is reflexive, symmetric and transitive, and $f \sim g$ if and only if $f - g = \operatorname{o}(g)$ .
  Uses: 6. Algebra of limits. Proved in: Equivalent infinitesimals and infinitely large functions.
9. Substitution of equivalents. If $f \sim f_1$ and $g \sim g_1$ at $P$ , then $\frac{f h}{g}$ has a limit, finite or infinite, if and only if $\frac{f_1 h}{g_1}$ has one, and both coincide; the theorem holds for factors, not for summands.
  Uses: 6. Algebra of limits, 18. Rules with infinite limits, 30. Comparison of functions. Proved in: Equivalent infinitesimals and infinitely large functions.
10. Sum of equivalents of the same sign. If $f_1 \sim g_1$ and $f_2 \sim g_2$ at $P$ , and $g_1$ and $g_2$ are both positive, or both negative, near $P$ , then $f_1 + f_2 \sim g_1 + g_2$ .
  Uses: 4. Three lemmas, 10. Squeeze theorem, 30. Comparison of functions. Rests on: C (Bound criterion). Proved in: Equivalent infinitesimals and infinitely large functions.
11. Table of equivalents. At $0$ , $\sin x$ , $\tan x$ , $\arctan x$ , $e^x - 1$ and $\ln(1 + x)$ are equivalent to $x$ , $1 - \cos x \sim \frac{x^2}{2}$ , $a^x - 1 \sim x \ln a$ , $\left(1 + x\right)^r - 1 \sim r x$ , and every polynomial is equivalent to its lowest-degree term; the equivalences hold with any function that tends to $0$ without vanishing in place of $x$ .
  Uses: 7. Polynomials, roots and powers, 12. Fundamental limit, 27. Standard limits. Rests on: E (Change of variable). Proved in: Equivalent infinitesimals and infinitely large functions.

Toward the derivative. The chain stops where the limit ceases to be the object of study and becomes a tool: the following unit defines the derivative as the limit of the difference quotient, obtains its rules from the algebra of limits and from composition, the derivatives of the elementary functions from the standard limits, and the intermediate value theorem from the preservation of sign and completeness.