Derivative Rule of Bernoulli-l'Hôpital

This lecture covers the Bernoulli's Rule and l'Hôpital's Rule for differentiable functions in a neighborhood of a point xo, where g(x) is not equal to 0 and g'(x) is not equal to 0. It explores various scenarios where the limits of f(x) and g(x) are finite or infinite, leading to specific outcomes. Examples are provided to illustrate the application of these rules in practice, emphasizing the importance of continuity and differentiability. The lecture concludes with a generalization of the Mean Value Theorem and a proof involving continuous and differentiable functions on a closed interval [a, b].