The friar and mathematician Marin Mersenne was Descartes’ great friend. It was through letters Mersenne that Descartes talked to the wise men of Europe, including Fermat. In the next generation, the Englishman John Wallis created the symbol $\infty$ for infinity in 1655.

Marin Mersenne (1588–1648), Descartes' friend and main correspondent.
Marin Mersenne (1588–1648), Descartes’ friend and main correspondent.Euler’s Eyes Collection
John Wallis (1616–1703), creator of the ∞ symbol.
John Wallis (1616–1703), creator of the ∞ symbol.Euler’s Eyes Collection

At school we learned a canon, a rule: you will never divide by zero. But it’s almost never explained why. Let’s build the thought, gradually reducing the divisor:

$$\begin{aligned} \frac{1}{0{,}1} &= 10 \\ \frac{1}{0{,}01} &= 100 \\ \frac{1}{0{,}001} &= 1\,000 \\ \frac{1}{0{,}0001} &= 10\,000 \\ \frac{1}{0{,}00001} &= 100\,000 \end{aligned}$$

The closer the divisor is to zero, the higher the result, without stopping. We say that the limit is infinite:

$$\lim_{x \to 0^+} \frac{1}{x} = +\infty \qquad\qquad \lim_{x \to 0^-} \frac{1}{x} = -\infty.$$

On the right, the result increases to $+\infty$; on the left, go down to $-\infty$. There is not a single number that could be the result of $1 \div 0$, and therefore division by zero is not defined.

Approach x to zero from the right and left: the two sides of the graph run in opposite directions.

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