The old texts on this site began the subject with a joke. Take two equal numbers, $a=b$, and do some innocent-looking algebra:

$$\begin{aligned} a&=b\\ a^2&=ab\\ a^2-b^2&=ab-b^2\\ (a+b)(a-b)&=b(a-b)\\ a+b&=b\\ 2b&=b\\ 2&=1. \end{aligned}$$

Where is the mistake? Going from the fourth line to the fifth, we divided both sides by $a-b$. But $a=b$, so $a-b=0$, and we divided by zero. The script hides the crime: $a-b$ looks like any number, and only someone who remembers the first line notices that it is zero. A disguised division by zero is the simplest form of what mathematicians call an indeterminate form, the famous $\frac00$.

The problem calculus left behind

Newton’s and Leibniz’s calculus lived on quantities that were at the same time zero and not zero. To differentiate $x^2$, Leibniz computed $\frac{(x+dx)^2-x^2}{dx}=2x+dx$ and then “neglected” the $dx$, as if it were zero. But if $dx$ were zero, you could not have divided by it two lines earlier. It worked, and nobody could explain why.

In 1734 the Irish bishop George Berkeley published a fierce attack, The Analyst, aimed at the “infidel mathematicians” who mocked religious faith while believing in far stranger things:

And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

— George Berkeley, The Analyst (1734)

Berkeley was right about the logic and wrong about the calculus. The calculations worked; what was missing was a language that said what they did.

Cauchy: approaching without arriving

The language came from France. In 1821 Augustin-Louis Cauchy published the Cours d’analyse of the École Polytechnique and defined the central idea in words:

When the values successively attributed to the same variable approach a fixed value indefinitely, so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.

— Augustin-Louis Cauchy, Cours d'analyse (1821)

The secret lies in “as little as one wishes”. The $dx$ never reaches zero: it stays arbitrarily close. A few decades later the German Karl Weierstrass replaced the sentence with letters: for every margin $\varepsilon>0$, however small, there is a distance $\delta>0$ that guarantees the approximation. This is the famous “epsilon-delta” definition, still the terror of first-year university students:

$$\begin{aligned}\lim_{x\to a}f(x)=L\iff{}&\forall\varepsilon>0\ \exists\delta>0:\\ &0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon.\end{aligned}$$

Notice the signs $\forall$ (for all) and $\exists$ (there exists), which we will look at closely in the text on logic: the script of reasoning put at the service of calculus. The infinitely small stopped being a thing and became a promise: give me any $\varepsilon$, and I will find the $\delta$.

lim, the arrow and what they hide

The notation also took time. In 1786 the Swiss Simon L’Huilier wrote the abbreviation lim., with a full stop, for “limit”. The little arrow under lim, $x\to a$, only spread at the start of the twentieth century, mainly through the book A Course of Pure Mathematics (1908) by the Englishman G. H. Hardy. The final script is compact and active:

$$\lim_{x\to2}\frac{x^2-4}{x-2}=\lim_{x\to2}\frac{(x+2)(x-2)}{x-2}=\lim_{x\to2}(x+2)=4.$$

The arrow says something the equals sign would not: $x$ approaches $2$ but is never $2$. That is why we may cancel $x-2$, which is never zero along the way, without committing the crime of the proof that $1=2$. The difference between the two calculations lies entirely in the arrow.

Exercise: cancel with the arrow

Compute $\displaystyle\lim_{x\to3}\frac{x^3-3x^2}{x-3}$. Factorising the numerator: $x^3-3x^2=x^2(x-3)$. Since $x\to3$ without ever being $3$, we may cancel:

$$\lim_{x\to3}\frac{x^2(x-3)}{x-3}=\lim_{x\to3}x^2=9.$$

Substituting directly would give $\frac00$, an indeterminate form. The arrow licenses the cancellation the equals sign would forbid.

Exercise: a relative of the number e

Compute $\displaystyle\lim_{x\to\infty}\left(1+\frac1x\right)^{2x}$. Rewriting with the rules for powers:

$$\left(1+\frac1x\right)^{2x}=\left[\left(1+\frac1x\right)^{x}\right]^2\ \longrightarrow\ e^2.$$

The script lets you see the familiar limit hidden inside the new one.

Sources: Judith V. Grabiner, The Origins of Cauchy’s Rigorous Calculus (1981) · Florian Cajori, A History of Mathematical Notations · Augustin-Louis Cauchy, MacTutor · George Berkeley, MacTutor.

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