A proof of the L'H\^opital's rule

I. B. Kazakov

Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 49-69

Résumé

In this paper a new proof of the L'Hôpital's rule proposed for calculus lecturers is presented. The according theorem is formulated and proved for the six types of limit: $x \to a$, $x \to a + 0$, $x \to a - 0$, $x \to +\infty$, $x \to -\infty$, $x \to +\infty$, for the two indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$ and also for four values of limit $A \in (-\infty, +\infty)$, $A = -\infty$, $A = +\infty$, $A = \infty$. Thus, the theorem covers $6 * 2 * 4 = 48$ cases of the L'Hôpital's rule. The presented proof of the theorem differs from the traditional ones by using not only the Cachy definition of limit a function but also the Heine one. The single partial limit theorem is used as the important auxiliary statement allowing to apply the Heine definition of limit. This statement also allows to apply arithmetic properties of sequence limits to the proof of the indeterminate form $\frac{\infty}{\infty}$ and the limit $x \to a + 0$, i.e. for the case where the most significant simplification is achieved.

Détail

Keywords: the L'Hôpital's rule, partial limits, Heine definition of a limit of a function, calculus for the first-year students.