Limits and continuity: introduction

Summary

This lesson opens the unit on limits and continuity. It presents the path through the unit, which starts from the epsilon-delta definition of the limit and arrives, through lessons that reuse the earlier proofs instead of repeating them, at the continuity of the elementary functions and the comparison of infinitesimals; it reconstructs the history of the notion of limit, from Zeno’s paradoxes and the method of exhaustion to Weierstrass’s definition, by way of the infinitesimals of Newton and Leibniz and Berkeley’s critique; it formulates the problem that gave rise to it, which is to give meaning to a quotient that becomes $\frac{0}{0}$ , and examines why neither the idea of approaching nor a table of values is enough to decide a limit. It ends by pointing out where limits and continuity are used in the rest of the course, in other courses on the site and in science and technology.

Learning objectives

Describe the path through the unit and explain what each lesson proves and which earlier results it draws on.
Explain why instantaneous velocity, the tangent and area require the notion of limit, and why the intuitive idea of approaching and tables of values are not enough to define it or to compute it.
Place in their historical context the stages of the notion of limit, from the method of exhaustion to the epsilon-delta definition, and identify the problem or the critique that drove each one.
Point out where limits and continuity come into play in the rest of the course, in other courses and in scientific and technical applications.

Entry assessment

Before working through the unit, it is advisable to measure the starting point. This self-assessment is not graded: it serves to know where to pause. The first ten questions cover, in order, the ten content lessons of the unit, one each, and measure a skill of computation or of reasoning, not a statement to be remembered; the same skills are measured again at the end, in the synthesis lesson, with other data. The last six check what this unit takes as known from Algebra and Geometry I and II: absolute value, the supremum, composition of functions, sign charts, the inequality between the sine, the arc and the tangent, and the fundamental inequality of the exponential.

The contents of the unit

In Algebra and Geometry I, the lesson on rational functions used vertical asymptotes and asymptotes at infinity, the continuity of those functions and the intermediate value theorem as admitted facts, without proof, because they required a definition of limit that was not yet available; and Algebra and Geometry II admitted in the same way the continuity of the trigonometric functions and of their inverses. This unit gives that definition and proves the admitted facts one by one, except the intermediate value theorem, which will be proved in the following unit. The proofs are arranged so as to be reused: a few are done with epsilon and delta, and the others cite the earlier results instead of repeating the argument. The lessons follow this order:

The limit of a function turns the idea of approaching into the epsilon-delta definition, specifies with accumulation points where it makes sense, writes the definition and its negation as quantified statements, and proves that the limit is unique, that it depends only on the nearby values and that it forces the function to stay bounded and to keep its sign near the point.
The algebra of limits reduces the computation rules to three lemmas proved with epsilon and delta, the main one of which is the bound criterion, and deduces from them, with a lemma according to which restricting the domain does not lose the limit, the limits of the sum, the product, the reciprocal and the quotient, those of polynomials, rational functions, roots and powers with rational exponents, and the theorem on the limit of a composition, which justifies the change of variable.
The squeeze theorem proves, with the bound criterion, that limits preserve non-strict inequalities, the theorem itself and the limit of the product of a bounded function and one that tends to zero; with these it obtains the limits of the sine and the cosine and the fundamental limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$ , from which the other trigonometric limits are deduced.
One-sided limits are defined as limits of a restriction, so that they inherit all the earlier theorems; the lesson proves that the limit exists if and only if the two one-sided limits exist and coincide, and that a bounded monotonic function has a one-sided limit equal to the supremum or the infimum of its values, the first theorem that guarantees that a limit exists without knowing its value.
The limit at infinity defines the limit when the variable grows without bound and proves, with the inversion lemma, that it is equivalent to a one-sided limit at $0$ , so that all the earlier theorems also hold at infinity; with them it computes the limits of powers and of rational functions.
Infinite limits are reduced, with the reciprocal lemma, to limits equal to zero; from this and from comparison come the rules for the sum, the product and the quotient, and the lesson shows that indeterminate forms can give any result, and proves the theorem on unbounded monotonic functions and the one on composition with infinite limits.
Asymptotes are defined with one-sided limits, limits at infinity and infinite limits; the lesson proves the formulas for the slope and the intercept of the oblique asymptote, the uniqueness of the non-vertical asymptote at each end and the facts about rational functions admitted in Algebra and Geometry, and proposes a method for sketching functions without derivatives.
Continuity is defined with the usual challenge and response, without excluding the point; the lesson proves that, at an accumulation point, it is equivalent to the limit existing and being equal to the value, classifies discontinuities as removable, jump and essential, inherits from the algebra of limits the continuity of the operations, and proves that the composition of continuous functions is continuous, so that the limit passes inside a continuous function.
The elementary functions turn out to be continuous by a single argument, that a monotonic function whose image is an interval cannot jump; with that continuity and the fundamental inequality of the exponential, the lesson proves the standard limits of the exponential and of the logarithm, obtains the number $e$ as a limit, resolves the form $1^\infty$ and compares the exponential, powers and the logarithm at infinity.
Equivalent infinitesimals and infinitely large functions are compared with little $\operatorname{o}$ , big $\operatorname{O}$ , the same order and equivalence; the lesson proves that, in a product or a quotient, each factor can be replaced by an equivalent one, but not in a sum, except when the equivalents of the terms have the same sign, and gathers the limits of the earlier lessons into a table of equivalents at zero.

The unit ends with the lesson of synthesis and challenges, which gathers what was achieved into a deductive chain, repeats in its exit assessment the skills of the entry assessment and proposes ten problems of greater difficulty.

The historical background

The logical order of the unit does not coincide with the historical one either. For more than a century, calculus was practiced successfully without a definition of limit; continuity was rigorously defined before the limit in its present form, and the epsilon-delta definition came at the end of the process, when calculus had already solved the problems for which it was born. It is worth going through that history, because the objections it had to overcome are the same ones that students usually raise.

The infinite among the Greeks

Zeno of Elea, who lived approximately between 490 and 425 before the common era, put forward arguments against the possibility of motion that are known above all through Aristotle’s Physics. The dichotomy argument holds that, to travel along a path, one must first reach its midpoint, before that the midpoint of the half, and so on without end, so that motion could never begin. The paradox raises, without resolving it, the question of whether infinitely many magnitudes can add up to a finite magnitude, as in $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1$ .

The Greek mathematicians sidestepped that question with the method of exhaustion, to which Eudoxus of Cnidus gave rigorous form around 370 before the common era: to prove that an area equals a certain quantity, one inscribes in the figure polygons whose difference from it can be made smaller than any given magnitude, and one proves by reduction to absurdity that the area can be neither greater nor smaller than that quantity. Around 225 before the common era, Archimedes proved in this way that the area of the region bounded by a parabola and one of its chords is $\frac{4}{3}$ of the area of the inscribed triangle with the same base and the same vertex, by adding up the triangles added at each stage, whose areas form the progression $1 + \frac{1}{4} + \frac{1}{16} + \cdots$ . The scheme of the argument, a quantity that can be made smaller than any given magnitude, is already that of the challenge and the response.

The infinitesimals of Newton and Leibniz

In the seventeenth century, the problem of tangents and that of areas received a general method. Isaac Newton wrote in October 1666 a treatise on fluxions, which he did not publish, and in 1671 The Method of Fluxions and Infinite Series, which was published only in 1736. Gottfried Wilhelm Leibniz introduced the integral sign in a manuscript of October 29, 1675, and on November 21 he wrote for the first time an integral with the sign and the differential; he published his differential calculus in 1684 in the journal Acta Eruditorum of Leipzig, under the title Nova methodus pro maximis et minimis, and his integral calculus in the same journal in 1686. Both computed with increments that are treated as different from zero in order to divide by them and as zero in order to discard them at the end.

In 1734, George Berkeley published The Analyst, a critique of the foundations of Newton’s calculus. He maintained that the evanescent increments were neither finite quantities, nor infinitely small ones, nor nothing, that the calculus changed its hypothesis halfway through the reasoning, and he asked whether they might not be called ghosts of departed quantities. The results were correct; what Berkeley showed is that the reasoning that justified them was not. The example below reproduces the computation he criticized and shows how the definition of limit resolves it.

ExampleThe quotient Berkeley criticized

Consider the slope of the tangent to the parabola $y = x^2$ at the point $(1, 1)$ . The figure draws the secants that join that point with another point of the parabola ever closer to it and, afterwards, the quotient as a function of the increment, with the hole it leaves at $h = 0$ .

Proof

$q(h) = \frac{(1 + h)^2 - 1^2}{h}$
The secant that joins $(1, 1)$ with the point with abscissa $1 + h$ , where $h \neq 0$ , has as its slope the difference quotient.
$h \neq 0 \rightarrow \dato{q}{q(h) = 2 + h}$
For $h \neq 0$ , the numerator equals $2h + h^2$ , and the factor $h$ cancels. As $h$ shrinks, the secants become less and less steep.
$q(0) = \frac{0}{0}, \qquad h \neq 0 \rightarrow q(h) \neq 2$
Here is Berkeley’s objection: if $h = 0$ , the quotient is $\frac{0}{0}$ and cannot be simplified; if $h \neq 0$ , the result is not $2$ . The calculus of infinitesimals used $h \neq 0$ to divide and $h = 0$ to discard the term $h$ .
$0 < \abs{h} < \varepsilon \rightarrow \abs{q(h) - 2} = \resaltar{\abs{h}} < \varepsilon$
The definition of limit removes the objection: it does not examine $h = 0$ , nor does it require $q(h)$ to equal $2$ , but only that it be as close to $2$ as we like. The challenge $\varepsilon$ is met with the response $\delta = \varepsilon$ .
$\lim_{h \to 0} \frac{(1 + h)^2 - 1}{h} = \resaltar{2}, \qquad y = 2x - 1$
Therefore, the limit is $2$ , the slope of the tangent, and no quantity has to be at once zero and nonzero. The line with slope $2$ that passes through $(1, 1)$ is the tangent.

The limit as foundation

The way out that the example pointed to gradually made its way over a century. Jean le Rond d’Alembert, in the article “Différentiel” of the fourth volume of the Encyclopédie, written in 1754, proposed to found calculus on the theory of limits and defined the derivative as the limit of a quotient of increments. In 1817, Bernard Bolzano published a purely analytic proof of the intermediate value theorem, with the declared purpose of freeing calculus from the notion of the infinitesimal, and used in it a notion of continuity very close to the present one; his work was little known during his lifetime. In 1821, Augustin-Louis Cauchy published his Cours d’analyse, written for the students of the École polytechnique in Paris, which placed limits at the base of calculus and defined a continuous function as one in which an infinitely small increment of the variable produces an infinitely small increment of the function.

At the same time the notion of function was changing. For Euler, in 1748, a function was an analytic expression; Peter Gustav Lejeune Dirichlet accepted in 1837 a much broader notion, in which the values may follow one another in any manner, and to him we owe the example of the function that equals $0$ at the rationals and $1$ at the irrationals, discontinuous at every point. With such general functions, geometric intuition ceased to be a reliable guide, and definitions had to speak only of numbers and inequalities.

Weierstrass’s definition

Karl Weierstrass, professor at the University of Berlin from 1856, first addressed the foundations of analysis in his course of introduction to analysis of 1859 and 1860. In his lectures, known above all through his students’ notes, the definition of the limit and of continuity was written with epsilon and delta, in the form this course uses: without motion, without infinitesimals, as a quantified statement about real numbers. In 1872, the same year in which Richard Dedekind and Georg Cantor published their constructions of the real numbers, which prove completeness, Weierstrass presented to the Berlin Academy of Sciences a continuous function that has no derivative at any point, proof that the intuition of a continuous curve is not enough to reason about it.

The notation with which the last lesson of the unit compares functions is later. Paul Bachmann introduced the symbol $\operatorname{O}$ in 1894, in Die analytische Zahlentheorie, the second volume of his treatise on number theory, and Edmund Landau, who adopted it, introduced the symbol $\operatorname{o}$ in 1909, in his treatise on the distribution of prime numbers.

The problem that gave rise to it

Three ancient problems led to calculus, and all three have the same structure. The velocity of a moving body at an instant ought to be the quotient of the distance traveled and the time elapsed when the time elapsed is zero; the slope of the tangent to a curve ought to be that of a secant whose two points coincide; the area of a curved figure ought to be the sum of infinitely many rectangles of zero width. In all three cases, direct substitution produces $\frac{0}{0}$ or a meaningless sum, and the correct result is what the approximations approach. The first two lead to the derivative, which is the subject of the following unit; the third, to the integral, which is studied in Calculus II.

However, “what the approximations approach” is not yet a definition, for three reasons. The first is that approaching is not enough: the lesson on the limit of functions of a real variable shows that the values of $\frac{x^2 - 1}{x - 1}$ approach $3$ when $x$ goes from $0.9$ to $0.99$ , even though the limit is $2$ . The second is that a table examines finitely many points: the same lesson shows that $\sin\frac{\pi}{x}$ takes the value $0$ at every point of the form $\frac{1}{n}$ and yet has no limit at $0$ . The third is that the values in the table may be computed wrongly, as the example below shows.

ExampleA misleading table

Consider the function $q(x) = \frac{\sqrt{x^2 + 9} - 3}{x^2}$ , defined for $x \neq 0$ . The figure draws its graph, with the hole at $x = 0$ , the values of a table and the value a computer gives very close to $0$ ; afterwards, the bound that decides the limit.

Proof

$q(1) \approx 0.1623, \qquad q\left(10^{-1}\right) \approx 0.16662, \qquad q\left(10^{-2}\right) \approx 0.1666662$
The values at $1$ , $10^{-1}$ and $10^{-2}$ seem to approach $0.1666\ldots$ , that is, $\frac{1}{6}$ .
$9 + 10^{-16} \approx 9 \rightarrow q\left(10^{-8}\right) \approx \resaltar{0}$
But at $x = 10^{-8}$ , a program that computes with the usual double-precision floating point obtains $0$ : the sum $9 + 10^{-16}$ is rounded to $9$ , and the subtraction in the numerator gives exactly $0$ . The table alone does not decide between $\frac{1}{6}$ and $0$ .
$q(x) = \frac{\left(x^2 + 9\right) - 9}{x^2\left(\sqrt{x^2 + 9} + 3\right)} = \dato{r}{\frac{1}{\sqrt{x^2 + 9} + 3}}$
For $x \neq 0$ , we multiply and divide by the conjugate $\sqrt{x^2 + 9} + 3$ ; the numerator becomes $x^2$ , which cancels.
$\abs{q(x) - \frac{1}{6}} = \frac{x^2}{6\left(\sqrt{x^2 + 9} + 3\right)^2} \leq \resaltar{\frac{x^2}{216}}$
The distance to $\frac{1}{6}$ is written with the same conjugate, and it is bounded because $\sqrt{x^2 + 9} \geq 3$ , so that the square of $\sqrt{x^2 + 9} + 3$ is at least $36$ .
$0 < \abs{x} < \sqrt{216\varepsilon} \rightarrow \abs{q(x) - \frac{1}{6}} < \varepsilon, \qquad \lim_{x \to 0} q(x) = \resaltar{\frac{1}{6}}$
The challenge $\varepsilon$ is met with the response $\delta = \sqrt{216\varepsilon}$ , and the limit is $\frac{1}{6}$ . The computer’s $0$ is a rounding error: the bound guarantees that $q\left(10^{-8}\right)$ differs from $\frac{1}{6}$ by less than $10^{-18}$ .

The definition of limit responds to the three objections: it speaks of all nearby points at once, fixes with numbers what “as close as we like” means and is proved with inequalities, not with approximate computations. But the problem has another side, which is a matter not of computation but of existence. Many limits in the unit are proved after their value is known or guessed; others, by contrast, exist without our knowing how to compute them, and their existence rests on the completeness of the real numbers. The theorem on monotonic functions in the lesson on one-sided limits gives the limit as a supremum, and on it depends the continuity of the exponential, of the logarithm and of the other elementary functions. The lesson on limit and continuity proves that a function continuous and positive at a point is also positive at the points of its domain near it, and that preservation of sign, together with completeness, will yield in the following unit the intermediate value theorem. Without completeness none of this would hold: the function $x^2 - 2$ , considered only on the rationals, goes from negative to positive values without ever vanishing.

The relevance of limits and continuity

In the rest of the course

The whole course rests on this unit. The following unit, on derivatives, defines in its first lesson the derivative as the limit of a function, the difference quotient of Berkeley’s example, and obtains the differentiation rules from the algebra of limits; the derivatives of the sine, the exponential and the logarithm, in the lesson on the derivatives of the elementary functions, are the standard limits of this unit; the chain rule rests on the composition of continuous functions; the intermediate value theorem is proved with preservation of sign and completeness, and justifies the bisection method; and the derivative of the inverse, in the lesson on the derivative of the inverse and implicit differentiation, requires the continuity of the inverse. The third unit, on the applications of derivatives, begins with the Weierstrass theorem, according to which a function continuous on a closed and bounded interval attains its maximum and its minimum, and on it rest Rolle’s theorem and the mean value theorem, the search for maxima and minima and optimization problems. L’Hôpital’s rule resolves the indeterminate forms that this unit can only recognize, and the Newton-Raphson method, which closes the course, has its convergence proved with convexity and the supremum.

In other courses

In Calculus II, the integral is defined as the limit of sums of areas of rectangles, the problem of Archimedes; sequences and series, which this course does not study, are treated with definitions of the same kind, and the Taylor polynomial measures its error with the big $\operatorname{O}$ of the last lesson of the unit. In Calculus III, the epsilon-delta definition carries over to functions of several variables, with the distance in the plane instead of the absolute value, and the limit may depend on the path along which the point is approached. In Foundations of Mechanics, instantaneous velocity and acceleration are limits of quotients of increments. In Financial Mathematics, continuous compounding is the limit of $\left(1 + \frac{r}{x}\right)^{x}$ as $x$ tends to infinity, which equals $e^{r}$ by the lesson on standard limits. In Data Analysis with R, growth models are distinguished by their behavior at infinity, which is what the comparison of the exponential, powers and the logarithm describes. In Programming in C++ I, bisection approximates a root by relying on the intermediate value theorem, the cost of an algorithm is measured with the $\operatorname{O}$ notation of Bachmann and Landau, and floating-point numbers force us to distinguish the exact limit from the computed value, as in the misleading table.

In science and technology

Every magnitude that changes continuously, a position, a temperature, a concentration, a voltage, is described by continuous functions, and its instantaneous rates of change are limits. The approximations with which engineering and the experimental sciences work (the linearization of a phenomenon near an equilibrium point, the replacement of $\sin x$ by $x$ in small oscillations, the neglect of one term compared with another much larger one) are equivalences like those of the last lesson of the unit, and they hold under the same conditions. Asymptotes describe the saturation value of a process that settles down and the unbounded growth of one that does not. And numerical computation, which solves on a computer what has no closed formula, rests on the distinction in the example of the misleading table: an algorithm produces approximations, and only an error bound, proved with inequalities, says how close they come to the exact value.