This technique provides a method for evaluating limits involving indeterminate forms, such as 0/0 or /. It states that if the limit of the ratio of two functions, f(x) and g(x), as x approaches a certain value (c or infinity) results in an indeterminate form, then, provided certain conditions are met, the limit of the ratio of their derivatives, f'(x) and g'(x), will be equal to the original limit. For example, the limit of (sin x)/x as x approaches 0 is an indeterminate form (0/0). Applying this method, we find the limit of the derivatives, cos x/1, as x approaches 0, which equals 1.
This method is crucial for Advanced Placement Calculus students as it simplifies the evaluation of complex limits, eliminating the need for algebraic manipulation or other complex techniques. It offers a powerful tool for solving problems related to rates of change, areas, and volumes, concepts central to calculus. Developed by Guillaume de l’Hpital, a French mathematician, after whom it is named, this method was first published in his 1696 book, Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes, marking a significant advancement in the field of calculus.
