L'Hôpital's rule (Q2515077)

When \(f\) and \(g\) are differential real functions in an interval around a point \(\alpha\) and \(\lim_{x\to\alpha} {f'(x)\over g'(x)}\) exists, then \(\lim_{x\to\alpha} {f(x)\over g(x)}\) exists also and both these values coincide. This is L'Hôpital's rule. In this paper, interesting knowledge around it, due to Cauchy, Leibniz, Bernoulli, Lacroix, Lagrange in their work has been treated. Relatively new sources are mentioned. Connections over three centuries are elucidated, as well as in the educational vain around the mathematical objects of study, the notions, the methods. According to the authors, in respect to modern standards, the proofs of L'Hôpital's rule due to Lacroix and Cauchy are the most reliable ones.