A proof of the L’Hˆopital’s rule

Abstract

In this paper a new proof of the L’Hˆopital’s rule proposed for calculus lecturers is presented. The according theorem is formulated and proved for the six types of limit: 𝑥 → 𝑎, 𝑥 → 𝑎 + 0, 𝑥 → 𝑎 − 0, 𝑥 → +∞, 𝑥 → −∞, 𝑥 → +∞, for the two indeterminate forms 0/0 and ∞/∞ and also for four values of limit 𝐴 ∈ (−∞,+∞), 𝐴 = −∞, 𝐴 = +∞, 𝐴 = ∞. Thus, the theorem covers 6 * 2 * 4 = 48 cases of the L’Hˆopital’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 ∞/∞ and the limit 𝑥 → 𝑎 + 0, i.e. for the case where the most significant simplification is achieved.