A rule for removing indeterminacies of the form or by reducing the limit of the ratio of functions to the limit of the ratio of the derivatives of the functions in question. Thus, for the case when the real-valued functions and are defined in a punctured right neighbourhood of a point on the number axis, l'Hospital's rule has the form
Both in the case of an indeterminacy of the form , that is, when
and in the case , that is, when
l'Hospital's rule is valid under the conditions that and are differentiable on some interval , for all points , and that there is a finite or infinite limit of the ratio of the derivatives:
(in the case of an indeterminacy of the form , this limit, if it is infinite, can only be an infinity of definite sign); then the limit of the ratio of the functions exists and (*) holds. This assertion remains true, with natural changes, for the case of a left-sided and also a two-sided limit, and also when or .
In a practical search for limits of ratios of functions by means of l'Hospital's rule one must sometimes apply it several times in succession.
Under the assumptions made above, the existence of a limit of the ratio of derivatives is a sufficient condition for the existence of a limit of the ratio of the functions themselves, but it is not necessary.
|||G.F. l'Hospital, "Analyse des infiniment petits pour l'intellligence des lignes courbes" , Paris (1696)|
|||S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)|
The "rule" is probably due to Johann Bernoulli, who taught the marquis de l'Hospital mathematics.
|[a1]||G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian)|
|[a2]||W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108|
|[a3]||K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)|
L'Hospital rule. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=L%27Hospital_rule&oldid=14236